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{{quantum mechanics | cTopic = Background }}
The '''ladder paradox''' (or '''barn-pole paradox''') is a [[thought experiment]] in [[special relativity]]. It involves a ladder travelling horizontally and undergoing a [[length contraction]], the result of which being that it can fit into a much smaller garage. On the other hand, from the point of view of an observer moving with the ladder, it is the garage that is moving and the garage will be contracted to an even smaller size, therefore being unable to contain the ladder at all.  This apparent paradox results from the assumption of absolute simultaneity. In relativity, [[Relativity of simultaneity|simultaneity is relative]] to each observer and thus the ladder can fit into the garage in both instances.
The '''old quantum theory''' is a collection of results from the years 1900–1925 which predate modern [[quantum mechanics]]. The theory was never complete or self-consistent, but was a set of [[heuristic]] prescriptions which are now understood to be the first quantum corrections to [[classical mechanics]].<ref>{{cite book|last = ter Haar|first =D.|title =The Old Quantum Theory|publisher=Pergamon Press|year=1967|pages = 206|isbn = 0-08-012101-2}}</ref> The [[Bohr model]] was the focus of study, and [[Arnold Sommerfeld]]<ref>{{cite book|first=Arnold |last= Sommerfeld|title=Atombau und Spektrallinien'|  publisher=Friedrich Vieweg und Sohn|location=Braunschweig|year=1919|isbn=3-87144-484-7}}</ref> made a crucial contribution by quantizing the z-component of the [[angular momentum]], which in the old quantum era was called ''space quantization'' (Richtungsquantelung). This allowed the orbits of the electron to be ellipses instead of circles, and introduced the concept of [[quantum degeneracy]]. The theory would have correctly explained the [[Zeeman effect]], except for the issue of electron [[Spin (physics)|spin]].


The main tool was '''Bohr–Sommerfeld quantization''', a procedure for selecting out certain discrete set of states of a classical integrable motion as allowed states. These are like the allowed orbits of the Bohr model of the atom; the system can only be in one of these states and not in any states in between.
==Paradox==
[[File:Ladder Paradox Overview.svg|thumb|266px|left|Figure 1: An overview of the garage and the ladder at rest]]
The problem starts with a ladder and an accompanying garage that is too small to contain the ladder.  Through the relativistic effect of [[length contraction]], the ladder can be made to fit into the garage by running it into the garage at a high enough speed.
[[File:Ladder Paradox GarageFrame.svg|thumb|152px|right|Figure 2: In the garage frame, the ladder undergoes length contraction and will therefore fit into the garage.]]
However, both the ladder and garage occupy their own [[inertial frame of reference|inertial reference frames]], and thus both frames are equally valid frames to view the problem.  From the reference frame of the garage, it is the ladder that moves with a [[relative velocity]] and so it is the ladder that undergoes length contraction.
Conversely, through [[symmetry]], from the reference frame of the ladder it is the garage that is moving with a relative velocity and so it is the garage that undergoes a length contraction.  From this perspective, the garage is made even smaller and it is impossible to fit the ladder into the garage.[[File:Ladder Paradox LadderFrame.svg|thumb|150px|center|Figure 3: In the ladder frame, the garage undergoes length contraction and seems too small to contain the ladder.]]


== Basic principles ==
==Relative simultaneity==
The basic idea of the old quantum theory is that the motion in an atomic system is quantized, or discrete. The system obeys [[classical mechanics]] except that not every motion is allowed, only those motions which obey the ''old quantum condition'':
[[File:Ladder Paradox GarageScenario.svg|thumb|250px|left|Figure 4: Scenario in the garage frame: a length contracted ladder entering and exiting the garage]]
[[File:Ladder Paradox LadderScenario.svg|thumb|250px|Figure 5: Scenario in the ladder frame: a length contracted garage passing over the ladder]]
The solution to the apparent paradox lies in the fact that what one observer (e.g. the garage) considers as simultaneous does not correspond to what the other observer (e.g. the ladder) considers as simultaneous ([[Relativity of simultaneity|relative simultaneity]]).  A clear way of seeing this is to consider a garage with two doors that swing shut to contain the ladder and then open again to let the ladder out the other side.


:<math>
From the perspective of the garage, the length-contracted ladder is short enough to fit entirely inside. The instant the ladder is fully inside the garage, the front and back doors close simultaneously. Then, since the ladder is still moving at considerable speed, the front and back doors simultaneously open again to allow the ladder to exit.
\oint\limits_{H(p,q)=E} p_i \, dq_i = n_i h
</math>


where the <math>p_i</math> are the momenta of the system and the <math>q_i</math> are the corresponding coordinates. The quantum numbers <math>n_i</math> are ''integers'' and the integral is taken over one period of the motion at constant energy (as described by the [[hamiltonian mechanics|Hamiltonian]]). The integral is an area in phase space, which is a quantity called the action and is quantized in units of [[Planck constant|Planck's constant]]. For this reason, Planck's constant was often called the ''quantum of action''.
From the perspective of the ladder, the back door (right) closes and opens, then after the garage passes over the ladder, the front door (left) closes and opens.


In order for the old quantum condition to make sense, the classical motion must be separable, meaning that there are separate coordinates <math>q_i</math> in terms of which the motion is periodic. The periods of the different motions do not have to be the same, they can even be incommensurate, but there must be a set of coordinates where the motion decomposes in a multi-periodic way.
The situation is illustrated in the [[Minkowski diagram]] below. The diagram is in the rest frame of the garage. The vertical light-blue-shaded band shows the garage in space-time, the light-red band shows the ladder in space-time.  The x and t axes are the garage space and time axes, respectively, and x&prime; and t&prime; are the ladder space and time axes, respectively.  The ladder is moving at a velocity of <math>v=c\sqrt{1/2}</math> in the positive x direction, therefore <math>\gamma=\sqrt{2}</math>. (From the ladder's point of view, its speed in the x&prime; direction is the same.)


The motivation for the old quantum condition was the [[correspondence principle]], complemented by the physical observation that the quantities which are quantized must be [[adiabatic invariant]]s. Given Planck's quantization rule for the harmonic oscillator, either condition determines the correct classical quantity to quantize in a general system up to an additive constant.
Since light travels at very close to one foot per nanosecond, let’s work in these units, so that <math>c \approx 1 \mbox{ft/ns}</math>. The garage is a small one, G=10 feet long, while in the ladder frame, the ladder is L=12 feet long.  In the garage frame, the front of the ladder will hit the back of the garage at time <math>t_A=G/v\approx14.14\mbox{ ns}</math> (if <math>t_D = t_O = 0</math> is chosen as the reference point).  This is shown as event '''A''' on the diagram. All lines parallel to the garage x axis will be simultaneous according to the garage observer, so the dark blue line '''AB''' will be what the garage observer sees as the ladder at the time of event '''A'''. The ladder is contained inside the garage. However, from the point of view of the observer on the ladder, the dark red line '''AC''' is what the ladder observer sees as the ladder. The back of the ladder is outside the garage.


==Examples==
[[File:LadderParadox1 Minkowski.svg|center|thumb|325px|Figure 6: A Minkowski diagram of ladder paradox. The garage is shown in light blue, the ladder in light red. The diagram is in the rest frame of the garage, with x and t being the garage space and time axes, respectively. The ladder frame is for a person sitting on the front of the ladder, with x&prime; and t&prime; being the ladder space and time axes respectively. The x and x&prime; axes are each {{convert|5|ft|m}} long in their respective frames, and the t and t&prime; axes are each 5&nbsp;ns in duration.]]


=== Harmonic oscillator ===
==Resolution==
The simplest system in the old quantum theory is the [[harmonic oscillator]], whose [[Hamiltonian (quantum mechanics)|Hamiltonian]] is:


:<math>
[[File:ladder paradox contraction.png|thumb|left|143px|Figure 7: A ladder contracting under acceleration to fit into a length contracted garage]]
H= {p^2 \over 2m} + {m\omega^2 q^2\over 2}.
</math>


The level sets of ''H'' are the orbits, and the quantum condition is that the area enclosed by an orbit in phase space is an integer. It follows that the energy is quantized according to the Planck rule:
In the context of the paradox, when the ladder enters the garage and is contained within it, it must either continue out the back or come to a complete stop. When the ladder comes to a complete stop, it accelerates into the reference frame of the garage.  From the reference frame of the garage, all parts of the ladder come to a complete stop simultaneously, and thus all parts must accelerate simultaneously.


:<math>
From the reference frame of the ladder, it is the garage that is moving, and so in order to be stopped with respect to the garage, the ladder must accelerate into the reference frame of the garage.  All parts of the ladder cannot accelerate simultaneously because of relative simultaneity.  What happens is that each part of the ladder accelerates sequentially, front to back, until finally the back end of the ladder accelerates when it is within the garage, the result of which is that, from the reference frame of the ladder, the front parts undergo length contraction ''sequentially'' until the entire ladder fits into the garage.
E= n\hbar \omega,
\,</math>


a result which was known well before, and used to formulate the old quantum condition. This result differs by <math>\frac{1}{2}\hbar \omega</math> from the results found with the help of [[quantum mechanics]]. This constant is neglected in the derivation of the ''old quantum theory'', and its value can not be determined using it.
[[File:junk2.png|frame|center|Figure 8: A Minkowski diagram of the case where the ladder is stopped all along its length, simultaneously in the garage frame. When this occurs, the garage frame sees the ladder as AB, but the ladder frame sees the ladder as AC. When the back of the ladder enters the garage at point D, it has not yet felt the effects of the acceleration of its front end. At this time, according to someone at rest with respect to the back of the ladder, the front of the ladder will be at point E and will see the ladder as DE. It is seen that this length in the ladder frame is not the same as CA, the rest length of the ladder before the deceleration.]]


The thermal properties of a quantized oscillator may be found by averaging the energy in each of the discrete states assuming that they are occupied with a [[Boltzmann weight]]:
==Ladder paradox and transmission of force==


:<math>
[[File:Junk1.png|frame|right|Figure 1: A Minkowski diagram of the case where the ladder is stopped by impact with the back wall of the garage. The impact is event A. At impact, the garage frame sees the ladder as AB, but the ladder frame sees the ladder as AC. The ladder does not move out of the garage, so its front end now goes directly upward, through point E. The back of the ladder will not change its trajectory in space-time until it feels the effects of the impact. The effect of the impact can propagate outward from A no faster than the speed of light, so the back of the ladder will never feel the effects of the impact until point F or later, at which time the ladder is well within the garage in both frames. Note that when the diagram is drawn in the frame of the ladder, the speed of light is the same, but the ladder is longer, so it takes more time for the force to reach the back end; this gives enough time for the back of the ladder to move inside the garage.]]
U = {\sum_n \hbar\omega n e^{-\beta n\hbar\omega} \over \sum_n e^{-\beta n \hbar\omega}} = {\hbar \omega e^{-\beta\hbar\omega} \over 1 - e^{-\beta\hbar\omega}},\;\;\;{\rm where}\;\;\beta = \frac{1}{kT},
</math>


''kT'' is [[Boltzmann constant]] times the [[Thermodynamic temperature|absolute temperature]], which is the temperature as measured in more natural units of energy. The  quantity <math>\beta</math> is more fundamental in thermodynamics than the temperature, because it is the [[thermodynamic potential]] associated to the energy.
What if the back door (the door the ladder exits out of) is closed permanently and does not open? Suppose that the door is so solid that the ladder will not penetrate it when it collides, so it must stop. Then, as in the scenario described above, in the frame of reference of the garage, there is a moment when the ladder is completely within the garage (i.e. the back of the ladder is inside the front door), before it collides with the back door and stops. However, from the frame of reference of the ladder, the ladder is too big to fit in the garage, so by the time it collides with the back door and stops, the back of the ladder still has not reached the front door. This seems to be a paradox. The question is, does the back of the ladder cross the front door or not?


From this expression, it is easy to see that for large values of <math>\beta</math>, for very low temperatures, the average energy U in the Harmonic oscillator approaches zero very quickly, exponentially fast. The reason is that kT is the typical energy of random motion at temperature T, and when this is smaller than <math>\scriptstyle \hbar\omega</math>, there is not enough energy to give the oscillator even one quantum of energy. So the oscillator stays in its ground state, storing next to no energy at all.
The difficulty arises mostly from the assumption that the ladder is rigid (i.e. maintains the same shape). Ladders seem pretty rigid in everyday life. But being rigid requires that it can transfer force at infinite speed (i.e. when you push one end the other end must react immediately, otherwise the ladder will deform). This contradicts special relativity, which states that information can only travel at most the speed of light (which is too fast for us to notice in real life, but is significant in the ladder scenario). So objects cannot be perfectly rigid under special relativity.


This means that at very cold temperatures, the change in energy with respect to beta, or equivalently the change in energy with respect to temperature, is also exponentially small. The change in energy with respect to temperature is the [[specific heat]], so the specific heat is exponentially small at low temperatures, going to zero like
In this case, by the time the front of the ladder collides with the back door, the back of the ladder does not know it yet, so it keeps moving forwards (and the ladder "compresses"). In both the frame of the garage and the inertial frame of the ladder, the back end keeps moving at the time of the collision, until at least the point where the back of the ladder comes into the light cone of the collision (i.e. a point where force moving backwards at the speed of light from the point of the collision will reach it). At this point the ladder is actually shorter than the original contracted length, so the back end is well inside the garage. Calculations in both frames of reference will show this to be the case.


::<math> \exp(-\hbar\omega/kT) </math>
What happens after the force reaches the back of the ladder (the "green" zone in the diagram) is not specified. Depending on the physics, the ladder could break into a million pieces; or, if it were sufficiently elastic, it could re-expand to its original length and push the back end out of the garage.


At small values of <math>\beta</math>, at high temperatures, the average energy U is equal to <math>1/\beta = kT</math>. This reproduces the [[equipartition theorem]] of classical thermodynamics: every harmonic oscillator at temperature ''T'' has energy ''kT'' on average. This means that the specific heat of an oscillator is constant in classical mechanics and equal to&nbsp;''k''. For a collection of atoms connected by springs, a reasonable model of a solid, the total specific heat is equal to the total number of oscillators times&nbsp;''k''. There are overall three oscillators for each atom, corresponding to the three possible directions of independent oscillations in three dimensions. So the specific heat of a classical solid is always 3k per atom, or in chemistry units, 3R per [[Mole (unit)|mole]] of atoms.
<br style="clear:both;" />


Monatomic solids at room temperatures have approximately the same specific heat of 3k per atom, but at low temperatures they don't. The specific heat is smaller at colder temperatures, and it goes to zero at absolute zero. This is true for all material systems, and this observation is called the [[third law of thermodynamics]]. Classical mechanics cannot explain the third law, because in classical mechanics the specific heat is independent of the temperature.
==Man falling into grate variation==


This contradiction between classical mechanics and the specific heat of cold materials was noted by [[James Clerk Maxwell]] in the 19th century, and remained a deep puzzle for those who advocated an atomic theory of matter. Einstein resolved this problem in 1906 by proposing that atomic motion is quantized. This was the first application of quantum theory to mechanical systems. A short while later, [[Peter Debye]] gave a quantitative theory of solid specific heats in terms of quantized oscillators with various frequencies (see [[Einstein solid]] and [[Debye model]]).
[[File:ladder paradox grate variation.PNG|thumb|300px|A man (represented by a segmented rod) falling into a grate]]


=== One-dimensional potential: U=0 ===
This [[physical paradox|paradox]] was originally proposed and solved by [[Wolfgang Rindler]]<ref>{{Cite journal|author=Rindler, Wolfgang|title=Length Contraction Paradox|journal=American Journal of Physics|year=1961|volume=29|issue=6|pages=365–366|doi=10.1119/1.1937789|bibcode = 1961AmJPh..29..365R }}</ref> and involved a fast walking man, represented by a rod, falling into a grate.<ref>{{cite book|author=Edwin F. Taylor, John Archibald Wheeler|title=Spacetime Physics: Introduction to Special Relativity|year=1992|publisher=W. H. Freeman|location=New York|ISBN=0-7167-2327-1|pages=116}}</ref> It is assumed that the rod is entirely over the grate in the grate frame of reference before the downward acceleration begins simultaneously and equally applied to each point in the rod.
One-dimensional problems are easy to solve. At any energy ''E'', the value of the momentum ''p'' is found from the conservation equation:


:<math>
From the perspective of the grate, the rod undergoes a length contraction and fits into the grate.  However, from the perspective of the rod, it is the ''grate'' undergoing a length contraction, through which it seems the rod is then too long to fall.
\sqrt{2m(E - V(q))} = p
</math>


which is integrated over all values of ''q'' between the classical ''turning points'', the places where the momentum vanishes. The integral is easiest for a ''particle in a box'' of length ''L'', where the quantum condition is:
In fact, the downward acceleration of the rod, which is simultaneous in the grate's frame of reference, is not simultaneous in the rod's frame of reference. In the rod's frame of reference, the bottom of the front of the rod is first accelerated downward (not shown in drawing), and as time goes by, more and more of the rod is subjected to the downward acceleration, until finally the back of the rod is accelerated downward. This results in a bending of the rod in the rod's frame of reference. It should be stressed that, since this bending occurs in the rod's rest frame, it is a true physical distortion of the rod which will cause stresses to occur in the rod.


:<math>
<br style="clear:both;" />
2\int_0^L p \, dq = nh
</math>


which gives the allowed momenta:
==Bar and ring paradox==
[[File:BarAndRing.svg|thumb|right|360 px|The diagram on the left illustrates a bar and a ring in the rest frame of the ring at the instant that their centers coincide. The bar is Lorentz-contracted and moving upward and to the right while the ring is stationary and uncontracted. The diagram on the right illustrates the situation at the same instant, but in the rest frame of the bar. The ring is now Lorentz-contracted and rotated with respect to the bar, and the bar is uncontracted. Again, the ring passes over the bar without touching it.]]
The above paradox is complicated: It involves non-inertial frames of reference since at one moment the man is walking horizontally, and a moment later he is falling downward. It involves a physical deformation of the man (or segmented rod), since the rod is bent in one frame of reference and straight in another. These aspects of the problem introduce complications involving the stiffness of the bar which tends to obscure the real nature of the "paradox". A very similar but simpler problem involving only inertial frames is the "bar and ring" paradox (Ferraro 2007) in which a bar which is slightly larger in length than the diameter of a ring is moving upward and to the right with its long axis horizontal, while the ring is stationary and the plane of the ring is also horizontal. If the motion of the bar is such that the center of the bar coincides with the center of the ring at some point in time, then the bar will be Lorentz-contracted due to the forward component of its motion, and it will pass through the ring. The paradox occurs when the problem is considered in the rest frame of the bar. The ring is now moving downward and to the left, and will be Lorentz-contracted along its horizontal length, while the bar will not be contracted at all. How can the bar pass through the ring?


:<math>
The resolution of the paradox again lies in the relativity of simultaneity (Ferraro 2007). The length of a physical object is defined as the distance between two ''simultaneous'' events occurring at each end of the body, and since simultaneity is relative, so is this length. This variability in length is just the Lorentz contraction. Similarly, a physical angle is defined as the angle  formed by three ''simultaneous'' events, and this angle will also be a relative quantity. In the above paradox, although the rod and the plane of the ring are parallel in the rest frame of the ring, they are not parallel in the rest frame of the rod. The uncontracted rod passes through the Lorentz-contracted ring because the plane of the ring is rotated relative to the rod by an amount sufficient to let the rod pass through.
p= {nh \over 2L}
</math>


and the energy levels
In mathematical terms, a [[Lorentz transformation]] can be separated into the product of a spatial rotation and a "proper" Lorentz transformation which involves no spatial rotation. The mathematical resolution of the bar and ring paradox is based on the fact that the product of two proper Lorentz transformations may produce a Lorentz transformation which is not proper, but rather includes a spatial rotation component.


:<math>
==See also==
E_n= {p^2 \over 2m} = {n^2 h^2 \over 8mL^2}
* [[Twin paradox]]
</math>
* [[Bell's spaceship paradox]]
* [[Ehrenfest paradox]]
* [[Physical paradox]]
* [[Supplee's paradox]]
* [[Relativity of simultaneity]]


=== One-dimensional potential: U=Fx ===
==References==
<references />
* {{cite journal | author=Wells, Willard H. | title=Length paradox in relativity | journal=American Journal of Physics | volume=29 | issue =12|pages=858-858|year=1961| doi=10.1119/1.1937641|bibcode = 1961AmJPh..29..858W }}


Another easy case to solve with the old quantum theory is a linear potential on the positive halfline, the constant confining force ''F'' binding a particle to an impenetrable wall. This case is much more difficult in the full quantum mechanical treatment, and unlike the other examples, the semiclassical answer here is not exact but approximate, becoming more accurate at large quantum numbers.
* {{cite journal | author=Shaw, R. | title=Length contraction paradox | journal=American Journal of Physics | year=1962| volume=30 | issue =1|pages=72-72 |doi=10.1119/1.1941907|bibcode = 1962AmJPh..30...72S }}


:<math>
* {{cite journal | author=Martins, Roberto De A. | title=Length paradox in relativity | journal=American Journal of Physics | volume=46 | issue =6|pages=667–670|year=1978| doi=10.1119/1.11227|bibcode = 1978AmJPh..46..667M }}
2 \int_0^{\frac{E}{F}} \sqrt{2m(E - Fx)}\ dx= n h
</math>


so that the quantum condition is
* {{cite journal | author=Sastry, G. P. | title=Is length contraction really paradoxical? | journal=American Journal of Physics | volume=55 | issue =10|pages=943–946|year=1987| doi=10.1119/1.14911|bibcode = 1987AmJPh..55..943S }}


:<math>
* {{cite journal | author=Grøn, Øyvind; Johannesen, Steinar | title=Computer simulation of Rindler's length contraction paradox | journal=European Journal of Physics | volume=14 | issue =3|pages=97–100 |year=1993| doi=10.1088/0143-0807/14/3/001|bibcode = 1993EJPh...14...97G }}
{4\over 3} \sqrt{2m}{ E^{3/2}\over F } = n h
</math>


which determines the energy levels,
* {{cite journal | author=van Lintel, Harald; Gruber, Christian | title=The rod and hole paradox re-examined | journal=European Journal of Physics | year=2005| volume=26 | issue =1|pages=19–23|doi=10.1088/0143-0807/26/1/003|bibcode = 2005EJPh...26...19V }}


:<math>
* {{cite journal | author=Iyer, Chandru; Prabhu, G. M. | title=Reversal in the time order of interactive events: the collision of inclined rods | journal=European Journal of Physics | volume=27 | issue =4|pages=819–824|year=2008|doi=10.1088/0143-0807/27/4/013|arxiv=0809.1721|bibcode = 2006EJPh...27..819I }}
E_n = \left({3nhF\over 4\sqrt{2m}} \right)^{2/3}
</math>


In the specific case F=mg, the particle is confined by the gravitational potential of the earth and the "wall" here is the surface of the earth.
* {{cite journal | author=Pierce, Evan | title=The lock and key paradox and the limits of rigidity in special relativity | journal=American Journal of Physics | volume=75 | issue =7|pages=610–614|year=2007| doi=10.1119/1.2711827|bibcode = 2007AmJPh..75..610P }}


=== One-dimensional potential: U=kx^2 ===
* {{cite journal | author=Iyer, Chandru; Prabhu, G. M. | title=Differing observations on the landing of the rod into the slot | journal=American Journal of Physics | volume=74 | issue =11|pages=998–1001|year=2008| doi=10.1119/1.2346686|arxiv=0809.1740|bibcode = 2006AmJPh..74..998I }}


=== Rotator ===
* {{cite journal | author=McGlynn, Enda; van Kampen, Paul | title=A note on linking electrical current, magnetic fields, charges and the pole in a barn paradox in special relativity | journal=European Journal of Physics | volume=29 | issue =6|pages=N63-N67|year=2008| doi=10.1088/0143-0807/29/6/N03|bibcode = 2008EJPh...29...63M }}
Another simple system is the rotator. A rotator consists of a mass ''M'' at the end of a massless rigid rod of length ''R'' and in two dimensions has the Lagrangian:
:<math>
L = {MR^2 \over 2} \dot\theta^2
</math>


which determines that the angular momentum ''J'' conjugate to <math>\theta</math>, the [[polar coordinates|polar angle]], <math>\scriptstyle  J = MR^2 \dot\theta</math>. The old quantum condition requires that ''J'' multiplied by the period of <math>\theta</math> is an integer multiple of Planck's constant:
==Further reading==
 
:<math>
2\pi J = n h
\,</math>
 
the angular momentum to be an integer multiple of <math>\scriptstyle \hbar</math>. In the [[Bohr model]], this restriction imposed on circular orbits was enough to determine the energy levels.
 
In three dimensions, a rigid rotator can be described by two angles — <math>\scriptstyle \theta</math> and <math>\scriptstyle \phi</math>, where <math>\scriptstyle \theta</math> is the inclination relative to an arbitrarily chosen ''z''-axis while <math>\scriptstyle \phi</math> is the rotator angle in the projection to the ''x''–''y'' plane. The kinetic energy is again the only contribution to the Lagrangian:
 
:<math>
L = {MR^2\over 2} \dot\theta^2 + {MR^2\over 2} (\sin(\theta)\dot\phi)^2
\,</math>
 
And the conjugate momenta are <math>\scriptstyle p_\theta = \dot\theta</math> and <math>\scriptstyle p_\phi=\sin(\theta)^2 \dot\phi</math>. The equation of motion for <math>\scriptstyle \phi</math> is trivial: <math>\scriptstyle p_\phi</math> is a constant:
 
:<math>
p_\phi = l_\phi
\,</math>
 
which is the ''z''-component of the angular momentum.  The quantum condition demands that the integral of the constant <math>\scriptstyle l_\phi</math> as <math>\scriptstyle \phi</math> varies from 0 to <math>2\pi</math> is an integer multiple of ''h'':
 
:<math>
l_\phi = m \hbar
\,</math>


And ''m'' is called the [[magnetic quantum number]], because the ''z'' component of the angular momentum is the magnetic moment of the rotator along the ''z'' direction in the case where the particle at the end of the rotator is charged.
* Edwin F. Taylor and John Archibald Wheeler, '''Spacetime Physics (2nd ed)''' (Freeman, NY, 1992)
: - discusses various apparent SR paradoxes and their solutions


Since the three-dimensional rotator is rotating about an axis, the total angular momentum should be restricted in the same way as the two-dimensional rotator. The two quantum conditions restrict the total angular momentum and the ''z''-component of the angular momentum to be the integers ''l'',''m''. This condition is reproduced in modern quantum mechanics, but in the era of the old quantum theory it led to a paradox: how can the orientation of the angular momentum relative to the arbitrarily chosen ''z''-axis be quantized? This seems to pick out a direction in space.
*{{cite book | author=Rindler, Wolfgang | title=Relativity: Special, General and Cosmological | publisher=Oxford University Press | year=2001 | isbn =0-19-850836-0 }}


This phenomenon, the quantization of angular momentum about an axis, was given the name ''space quantization'', because it seemed incompatible with rotational invariance. In modern quantum mechanics, the angular momentum is quantized the same way, but the discrete states of definite angular momentum in any one orientation are [[quantum superposition]]s of the states in other orientations, so that the process of quantization does not pick out a preferred axis. For this reason, the name "space quantization" fell out of favor, and the same phenomenon is now called the quantization of angular momentum.
*{{cite book|last=Ferraro|first=Rafael|title=Einstein's space-time: an introduction to special and general relativity|publisher = [[Springer Science+Business Media|Springer]]|year = 2007|isbn = 978-0-387-69946-2|url=http://books.google.com/books?id=wa3CskhHaIgC&dq=relativity+bar+and+ring}}


=== Hydrogen atom ===
==External links==
The angular part of the Hydrogen atom is just the rotator, and gives the quantum numbers ''l'' and ''m''. The only remaining variable is the radial coordinate, which executes a periodic one-dimensional potential motion, which can be solved.
* [http://math.ucr.edu/~jdp/Relativity/SpecialRelativity.html Special Relativity Animations] ''from John de Pillis.''This  inter-active animated train-and-tunnel paradox is an analog of the pole (train) and barn (tunnel) paradox.


For a fixed value of the total angular momentum ''L'', the Hamiltonian for a classical Kepler problem is (the unit of mass and unit of energy redefined to absorb two constants):
[[Category:Physical paradoxes]]
 
[[Category:Special relativity]]
:<math>
[[Category:Thought experiments in physics]]
H= { p^2 \over 2 } + {l^2 \over 2 r^2 } - {1\over r}.
</math>
 
Fixing the energy to be (a negative) constant and solving for the radial momentum ''p'', the quantum condition integral is:
 
:<math>
2 \int \sqrt{2E - {l^2\over r^2} + { 2\over r}}\ dr=  k h
</math>
 
which is elementary, and gives a new quantum number ''k'' which determines the energy in combination with ''l''. The energy is:
 
:<math>
E= -{1 \over 2 (k + l)^2}
</math>
 
and it only depends on the sum of ''k'' and ''l'', which is the ''principal quantum number'' ''n''. Since ''k'' is positive, the allowed values of ''l'' for any given ''n'' are no bigger than ''n''. The energies reproduce those in the [[Bohr model]], except with the correct quantum mechanical multiplicities, with some ambiguity at the extreme values.
 
The semiclassical hydrogen atom is called the [[Arnold Sommerfeld|Sommerfeld]] model, and its orbits are ellipses of various sizes at discrete inclinations. The Sommerfeld model predicted that the magnetic moment of an atom measured along an axis will only take on discrete values, a result which seems to contradict rotational invariance but which was confirmed by the [[Stern–Gerlach experiment]].
'''Bohr–Sommerfeld theory''' is a part of the development of [[quantum mechanics]] and describes the possibility of atomic [[energy levels]] being split by a [[magnetic field]].
 
=== Relativistic orbit ===
[[Arnold Sommerfeld]] derived the relativistic solution of atomic energy levels.<ref>{{cite book | author = [[Arnold Sommerfeld]] | title = Atombau und Spektrallinien | publisher = Braunschweig | year = 1921}} [https://archive.org/stream/atombauundspekt00sommgoog#page/n541/mode/2up]</ref> We will start this derivation with the relativistic equation for energy in the [[electric potential]]
 
:<math>W={m_\mathrm{0} c^2} \left( \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} - 1 \right) - k \frac{Z e^2}{r}</math>
 
After substitution <math>u = \frac{1}{r}</math> we get
 
:<math>\frac{1}{\sqrt{1-\frac{v^2}{c^2}}} = 1 + \frac{W}{m_\mathrm{0} c^2} + k \frac{Z e^2}{m_\mathrm{0} c^2} u</math>
 
For momentum <math>p_\mathrm{r} = m \dot{r}</math>,  <math>p_\mathrm{\varphi} = m r^2 \dot{\varphi}</math> and their ratio <math>\frac{p_\mathrm{r}}{p_\mathrm{\varphi}} = - \frac{du}{d \varphi}</math> the equation of motion is (see [[Binet equation]])
 
:<math>\frac{d^2 u}{d \varphi ^2} = - \left( 1 - k^2 \frac{Z^2 e^4}{c^2 p_\mathrm{\varphi}^2} \right) u + \frac{m_\mathrm{0} kZe^2}{p_\mathrm{\varphi}^2} \left( 1+\frac{W}{m_\mathrm{0} c^2} \right) = - \omega_\mathrm{0}^2 u + K </math>
 
with solution
 
:<math> u = \frac{1}{r} = K + A \cos \omega_\mathrm{0} \varphi </math>
 
The angular shift of [[periapsis]] per revolution is given by
 
:<math> \varphi_\mathrm{s} = 2 \pi \left(\frac{1}{\omega_\mathrm{0}} - 1\right) \approx 4 \pi^3 k^2 \frac{Z^2 e^4}{c^2 n_\mathrm{\varphi}^2 h^2} </math>
 
With the quantum conditions
 
:<math> \oint p_\mathrm{\varphi} \, d \varphi = 2 \pi p_\mathrm{\varphi} = n_\mathrm{\varphi} h</math>
 
and
 
:<math> \oint p_\mathrm{r} \, d r = p_\mathrm{\varphi} \oint \left( \frac{1}{r} \frac{dr}{d \varphi} \right)^2 \, d \varphi = n_\mathrm{r} h</math>
 
we will obtain energies
 
:<math> \frac{W}{m_\mathrm{0} c^2} = \left( 1 + \frac{\alpha ^2 Z^2}{(n_\mathrm{r} + \sqrt{n_\mathrm{\varphi} ^2 - \alpha ^2 Z^2} )^2} \right) ^{-1/2} - 1</math>
 
where <math>\alpha</math> is the [[fine-structure constant]]. This solution (using [[Substitution (algebra)|substitution]]s for quantum numbers) is equivalent to the solution of the [[Dirac equation]].<ref>{{cite journal|author=Ya I Granovski |url=http://www.iop.org/EJ/article/1063-7869/47/5/L06/PHU_47_5_L06.pdf |title=Sommerfeld formula and Dirac's theory |journal=Physics-Uspekhi |volume=47 |issue=5 |pages=523&ndash;524 |year=2004|doi=10.1070/PU2004v047n05ABEH001885|bibcode = 2004PhyU...47..523G }}</ref> Nevertheless these solutions fails to predict the [[Lamb shift]]s.
 
== De Broglie waves ==
In 1905, Einstein noted that the entropy of the quantized electromagnetic field oscillators in a box is, for short wavelength, equal to the entropy of a gas of point particles in the same box. The number of point particles is equal to the number of quanta. Einstein concluded that the quanta could be treated as if they were localizable objects
(see<ref name=einsta>{{cite journal | last = Einstein | first = Albert | authorlink = Albert Einstein | title = Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt | journal = Annalen der Physik | volume = 17 | pages = 132–148 | year = 1905 | url = http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_132-148.pdf | accessdate = 2008-02-18|bibcode = 1905AnP...322..132E |doi = 10.1002/andp.19053220607 | issue = 6 }}</ref> page 139/140), particles of light, and named them [[photon]]s.
 
Einstein's theoretical argument was based on [[thermodynamics]], on counting the number of states, and so was not completely convincing. Nevertheless, he concluded that light had attributes of [[wave particle duality|both waves and particles]], more precisely that an electromagnetic standing wave with frequency <math>\omega</math> with the quantized energy:
 
:<math>
E = n\hbar\omega
\,</math>
 
should be thought of as consisting of n photons each with an energy <math>\scriptstyle \hbar\omega</math>. Einstein could not describe how the photons were related to the wave.
 
The photons have momentum as well as energy, and the momentum had to be <math>\scriptstyle \hbar k</math> where <math>k</math> is the wavenumber of the electromagnetic wave. This is required by relativity, because the momentum and energy form a [[four-vector]], as do the frequency and wave-number.
 
In 1924, as a PhD candidate, [[Louis de Broglie]] proposed a new interpretation of the quantum condition. He suggested that all matter, electrons as well as photons, are described by waves obeying the relations.
 
:<math>
p = \hbar k
</math>
 
or, expressed in terms of wavelength <math>\lambda</math> instead,
 
:<math>
p = {h \over \lambda}
</math>
 
He then noted that the quantum condition:
 
:<math>
\int p \, dx = \hbar \int k \, dx = 2\pi\hbar n
</math>
 
counts the change in phase for the wave as it travels along the classical orbit, and requires that it be an integer multiple of <math>2\pi</math>. Expressed in wavelengths, the number of wavelengths along a classical orbit must be an integer. This is the condition for constructive interference, and it explained the reason for quantized orbits—the matter waves make [[standing waves]] only at discrete frequencies, at discrete energies.
 
For example, for a particle confined in a box, a standing wave must fit an integer number of wavelengths between twice the distance between the walls. The condition becomes:
 
:<math>
n\lambda = 2L
\,</math>
 
so that the quantized momenta are:
 
:<math>
p = \frac{nh}{2L}</math>
 
reproducing the old quantum energy levels.
 
This development was given a more mathematical form by Einstein, who noted that the phase function for the waves: <math>\theta(J,x)</math> in a mechanical system should be identified with the solution to the [[Hamilton–Jacobi equation]], an equation which even [[William Rowan Hamilton|Hamilton]] considered to be the short-wavelength limit of wave mechanics.
 
These ideas led to the development of the [[Schrödinger equation]].
 
== Kramers transition matrix ==
The old quantum theory was formulated only for special mechanical systems which could be separated into action angle variables which were periodic. It did not deal with the emission and absorption of radiation. Nevertheless, [[Hendrik Kramers]] was able to find heuristics for describing how emission and absorption should be calculated.
 
Kramers suggested that the orbits of a quantum system should be Fourier analyzed, decomposed into harmonics at multiples of the orbit frequency:
 
:<math>
X_n(t) = \sum_{k=-\infty}^{\infty} e^{ik\omega t} X_{n;k}
</math>
 
The index ''n'' describes the quantum numbers of the orbit, it would be ''n''–''l''–''m'' in the Sommerfeld model. The frequency <math>\omega</math> is the angular frequency of the orbit <math>\scriptstyle 2\pi/T_n</math> while ''k'' is an index for the Fourier mode. Bohr had suggested that the ''k''-th harmonic of the classical motion correspond to the transition from level ''n'' to level ''n''−''k''.
 
Kramers proposed that the transition between states were analogous to classical emission of radiation, which happens at frequencies at multiples of the orbit frequencies. The rate of emission of radiation is proportional to  <math>|X_k|^2</math>, as it would be in classical mechanics. The description was approximate, since the Fourier components did not have frequencies that exactly match the energy spacings between levels.
 
This idea led to the development of [[matrix mechanics]].
 
== Limitations of the old quantum theory ==
The old quantum theory had some limitations:<ref>{{cite book |first=G.S. |last=Chaddha |title=Quantum Mechanics |publisher=New Age international |location=New Dehli |year=2006 |isbn=81-224-1465-6 |pages=8–9 |url=http://books.google.com/books?id=Bzj2JcPeAHAC&pg=PA9}}</ref>
* The old quantum theory provides no means to calculate the intensities of the spectral lines.
* It fails to explain the anomalous Zeeman effect (that is, where the spin of the electron cannot be neglected).
* It cannot quantize chaotic systems, i.e. dynamical systems in which trajectories are neither closed nor periodic and whose analytical form does not exist. This presents a problem for systems as simple as a 2-electron atom which is classically chaotic analogously to the famous gravitational [[three-body problem]].
However it can be used to describe atoms with more than one electron (e.g. Helium) and the Zeeman effect.<ref>{{cite journal|last=E.A. Solov’ev|title=Classical approach in atomic physics|journal= European Physical Journal D|year=2011|volume=65|pages=331–351|doi=10.1140/epjd/e2011-20261-6|arxiv = 1003.4387 |bibcode = 2011EPJD...65..331S|first1=E. A.|issue=3 }}</ref>
It was later proposed that the old quantum theory is in fact the [[WKB approximation|semi-classical approximation]] to the canonical quantum mechanics <ref>{{cite book
|author=[[Lev Landau|L.D. Landau]], [[Evgeny Lifshitz|E.M. Lifshitz]]
|year=1977
|title=Quantum Mechanics: Non-Relativistic Theory
|edition=3rd |volume=Vol. 3
|publisher=[[Pergamon Press]]
|isbn=978-0-08-020940-1
}}</ref> but its limitations are still under investigation.
 
== History ==
The old quantum theory was sparked by the 1900 work of [[Max Planck]] on the emission and absorption of light, and began in earnest after the work of [[Albert Einstein]] on the specific heats of solids. Einstein, followed by Debye, applied quantum principles to the motion of atoms, explaining the specific heat anomaly.
 
In 1913, [[Niels Bohr]] identified the [[correspondence principle]] and used it to formulate a [[Bohr Model|model]] of the [[hydrogen atom]] which explained the [[Atomic emission spectrum|line spectrum]]. In the next few years [[Arnold Sommerfeld]] extended the quantum rule to arbitrary integrable systems making use of the principle of [[adiabatic invariant|adiabatic invariance]] of the quantum numbers introduced by Lorentz and Einstein. Sommerfeld's model was much closer to the modern quantum mechanical picture than Bohr's.
 
Throughout the 1910s and well into the 1920s, many problems were attacked using the old quantum theory with mixed results. Molecular rotation and vibration spectra were understood and the electron's spin was discovered, leading to the confusion of half-integer quantum numbers. [[Max Planck]] introduced the [[zero point energy]] and [[Arnold Sommerfeld]] semiclassically quantized the relativistic hydrogen atom. [[Hendrik Kramers]] explained the [[Stark effect]]. [[Satyendra Nath Bose|Bose]] and Einstein gave the correct quantum statistics for photons.
 
Kramers gave a prescription for calculating transition probabilities between quantum states in terms of Fourier components of the motion, ideas which were extended in collaboration with [[Werner Heisenberg]] to a semiclassical matrix-like description of atomic transition probabilities. Heisenberg went on to reformulate all of quantum theory in terms of a version of these transition matrices, creating [[matrix mechanics]].
 
In 1924, [[Louis de Broglie]] introduced the wave theory of matter, which was extended to a semiclassical equation for matter waves by Albert Einstein a short time later. In 1926 [[Erwin Schrödinger]] found a completely quantum mechanical wave-equation, which reproduced all the successes of the old quantum theory without ambiguities and inconsistencies. Schrödinger's wave mechanics developed separately from matrix mechanics until Schrödinger and others proved that the two methods predicted the same experimental consequences. Paul Dirac later proved in 1926 that both methods can be obtained from a more general method called [[Transformation theory (quantum mechanics)|transformation theory]].
 
==References==
{{reflist}}
 
==Further reading==
* {{Cite book| editor-first=J. |editor-last= Thewlis|title=Encyclopaedic Dictionary of Physics| year=1962}}
* {{cite journal
| last = Pais
| first = Abraham
| title = Max Born's Statistical Interpretation of Quantum Mechanics
| journal = Science
| volume =218
| issue =4578
| pages =1193–8
| publisher =
| location =
| url =http://cds.cern.ch/record/141137/files/cer-000052203.pdf
| jstor =
| doi =10.1126/science.218.4578.1193
| year = 1982
| pmid = 17802457 |bibcode = 1982Sci...218.1193P }} Address to annual meeting of the Optical Society of America October 21, 1982 (Tucson AZ). Retrieved 2013-09-08.


{{DEFAULTSORT:Old Quantum Theory}}
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[[lt:Kopėčių paradoksas]]
[[hu:Létraparadoxon]]

Revision as of 10:40, 13 August 2014

The ladder paradox (or barn-pole paradox) is a thought experiment in special relativity. It involves a ladder travelling horizontally and undergoing a length contraction, the result of which being that it can fit into a much smaller garage. On the other hand, from the point of view of an observer moving with the ladder, it is the garage that is moving and the garage will be contracted to an even smaller size, therefore being unable to contain the ladder at all. This apparent paradox results from the assumption of absolute simultaneity. In relativity, simultaneity is relative to each observer and thus the ladder can fit into the garage in both instances.

Paradox

Figure 1: An overview of the garage and the ladder at rest

The problem starts with a ladder and an accompanying garage that is too small to contain the ladder. Through the relativistic effect of length contraction, the ladder can be made to fit into the garage by running it into the garage at a high enough speed.

Figure 2: In the garage frame, the ladder undergoes length contraction and will therefore fit into the garage.

However, both the ladder and garage occupy their own inertial reference frames, and thus both frames are equally valid frames to view the problem. From the reference frame of the garage, it is the ladder that moves with a relative velocity and so it is the ladder that undergoes length contraction.

Conversely, through symmetry, from the reference frame of the ladder it is the garage that is moving with a relative velocity and so it is the garage that undergoes a length contraction. From this perspective, the garage is made even smaller and it is impossible to fit the ladder into the garage.

Figure 3: In the ladder frame, the garage undergoes length contraction and seems too small to contain the ladder.

Relative simultaneity

Figure 4: Scenario in the garage frame: a length contracted ladder entering and exiting the garage
Figure 5: Scenario in the ladder frame: a length contracted garage passing over the ladder

The solution to the apparent paradox lies in the fact that what one observer (e.g. the garage) considers as simultaneous does not correspond to what the other observer (e.g. the ladder) considers as simultaneous (relative simultaneity). A clear way of seeing this is to consider a garage with two doors that swing shut to contain the ladder and then open again to let the ladder out the other side.

From the perspective of the garage, the length-contracted ladder is short enough to fit entirely inside. The instant the ladder is fully inside the garage, the front and back doors close simultaneously. Then, since the ladder is still moving at considerable speed, the front and back doors simultaneously open again to allow the ladder to exit.

From the perspective of the ladder, the back door (right) closes and opens, then after the garage passes over the ladder, the front door (left) closes and opens.

The situation is illustrated in the Minkowski diagram below. The diagram is in the rest frame of the garage. The vertical light-blue-shaded band shows the garage in space-time, the light-red band shows the ladder in space-time. The x and t axes are the garage space and time axes, respectively, and x′ and t′ are the ladder space and time axes, respectively. The ladder is moving at a velocity of in the positive x direction, therefore . (From the ladder's point of view, its speed in the x′ direction is the same.)

Since light travels at very close to one foot per nanosecond, let’s work in these units, so that . The garage is a small one, G=10 feet long, while in the ladder frame, the ladder is L=12 feet long. In the garage frame, the front of the ladder will hit the back of the garage at time (if is chosen as the reference point). This is shown as event A on the diagram. All lines parallel to the garage x axis will be simultaneous according to the garage observer, so the dark blue line AB will be what the garage observer sees as the ladder at the time of event A. The ladder is contained inside the garage. However, from the point of view of the observer on the ladder, the dark red line AC is what the ladder observer sees as the ladder. The back of the ladder is outside the garage.

Figure 6: A Minkowski diagram of ladder paradox. The garage is shown in light blue, the ladder in light red. The diagram is in the rest frame of the garage, with x and t being the garage space and time axes, respectively. The ladder frame is for a person sitting on the front of the ladder, with x′ and t′ being the ladder space and time axes respectively. The x and x′ axes are each Template:Convert long in their respective frames, and the t and t′ axes are each 5 ns in duration.

Resolution

Figure 7: A ladder contracting under acceleration to fit into a length contracted garage

In the context of the paradox, when the ladder enters the garage and is contained within it, it must either continue out the back or come to a complete stop. When the ladder comes to a complete stop, it accelerates into the reference frame of the garage. From the reference frame of the garage, all parts of the ladder come to a complete stop simultaneously, and thus all parts must accelerate simultaneously.

From the reference frame of the ladder, it is the garage that is moving, and so in order to be stopped with respect to the garage, the ladder must accelerate into the reference frame of the garage. All parts of the ladder cannot accelerate simultaneously because of relative simultaneity. What happens is that each part of the ladder accelerates sequentially, front to back, until finally the back end of the ladder accelerates when it is within the garage, the result of which is that, from the reference frame of the ladder, the front parts undergo length contraction sequentially until the entire ladder fits into the garage.

Figure 8: A Minkowski diagram of the case where the ladder is stopped all along its length, simultaneously in the garage frame. When this occurs, the garage frame sees the ladder as AB, but the ladder frame sees the ladder as AC. When the back of the ladder enters the garage at point D, it has not yet felt the effects of the acceleration of its front end. At this time, according to someone at rest with respect to the back of the ladder, the front of the ladder will be at point E and will see the ladder as DE. It is seen that this length in the ladder frame is not the same as CA, the rest length of the ladder before the deceleration.

Ladder paradox and transmission of force

Figure 1: A Minkowski diagram of the case where the ladder is stopped by impact with the back wall of the garage. The impact is event A. At impact, the garage frame sees the ladder as AB, but the ladder frame sees the ladder as AC. The ladder does not move out of the garage, so its front end now goes directly upward, through point E. The back of the ladder will not change its trajectory in space-time until it feels the effects of the impact. The effect of the impact can propagate outward from A no faster than the speed of light, so the back of the ladder will never feel the effects of the impact until point F or later, at which time the ladder is well within the garage in both frames. Note that when the diagram is drawn in the frame of the ladder, the speed of light is the same, but the ladder is longer, so it takes more time for the force to reach the back end; this gives enough time for the back of the ladder to move inside the garage.

What if the back door (the door the ladder exits out of) is closed permanently and does not open? Suppose that the door is so solid that the ladder will not penetrate it when it collides, so it must stop. Then, as in the scenario described above, in the frame of reference of the garage, there is a moment when the ladder is completely within the garage (i.e. the back of the ladder is inside the front door), before it collides with the back door and stops. However, from the frame of reference of the ladder, the ladder is too big to fit in the garage, so by the time it collides with the back door and stops, the back of the ladder still has not reached the front door. This seems to be a paradox. The question is, does the back of the ladder cross the front door or not?

The difficulty arises mostly from the assumption that the ladder is rigid (i.e. maintains the same shape). Ladders seem pretty rigid in everyday life. But being rigid requires that it can transfer force at infinite speed (i.e. when you push one end the other end must react immediately, otherwise the ladder will deform). This contradicts special relativity, which states that information can only travel at most the speed of light (which is too fast for us to notice in real life, but is significant in the ladder scenario). So objects cannot be perfectly rigid under special relativity.

In this case, by the time the front of the ladder collides with the back door, the back of the ladder does not know it yet, so it keeps moving forwards (and the ladder "compresses"). In both the frame of the garage and the inertial frame of the ladder, the back end keeps moving at the time of the collision, until at least the point where the back of the ladder comes into the light cone of the collision (i.e. a point where force moving backwards at the speed of light from the point of the collision will reach it). At this point the ladder is actually shorter than the original contracted length, so the back end is well inside the garage. Calculations in both frames of reference will show this to be the case.

What happens after the force reaches the back of the ladder (the "green" zone in the diagram) is not specified. Depending on the physics, the ladder could break into a million pieces; or, if it were sufficiently elastic, it could re-expand to its original length and push the back end out of the garage.


Man falling into grate variation

A man (represented by a segmented rod) falling into a grate

This paradox was originally proposed and solved by Wolfgang Rindler[1] and involved a fast walking man, represented by a rod, falling into a grate.[2] It is assumed that the rod is entirely over the grate in the grate frame of reference before the downward acceleration begins simultaneously and equally applied to each point in the rod.

From the perspective of the grate, the rod undergoes a length contraction and fits into the grate. However, from the perspective of the rod, it is the grate undergoing a length contraction, through which it seems the rod is then too long to fall.

In fact, the downward acceleration of the rod, which is simultaneous in the grate's frame of reference, is not simultaneous in the rod's frame of reference. In the rod's frame of reference, the bottom of the front of the rod is first accelerated downward (not shown in drawing), and as time goes by, more and more of the rod is subjected to the downward acceleration, until finally the back of the rod is accelerated downward. This results in a bending of the rod in the rod's frame of reference. It should be stressed that, since this bending occurs in the rod's rest frame, it is a true physical distortion of the rod which will cause stresses to occur in the rod.


Bar and ring paradox

The diagram on the left illustrates a bar and a ring in the rest frame of the ring at the instant that their centers coincide. The bar is Lorentz-contracted and moving upward and to the right while the ring is stationary and uncontracted. The diagram on the right illustrates the situation at the same instant, but in the rest frame of the bar. The ring is now Lorentz-contracted and rotated with respect to the bar, and the bar is uncontracted. Again, the ring passes over the bar without touching it.

The above paradox is complicated: It involves non-inertial frames of reference since at one moment the man is walking horizontally, and a moment later he is falling downward. It involves a physical deformation of the man (or segmented rod), since the rod is bent in one frame of reference and straight in another. These aspects of the problem introduce complications involving the stiffness of the bar which tends to obscure the real nature of the "paradox". A very similar but simpler problem involving only inertial frames is the "bar and ring" paradox (Ferraro 2007) in which a bar which is slightly larger in length than the diameter of a ring is moving upward and to the right with its long axis horizontal, while the ring is stationary and the plane of the ring is also horizontal. If the motion of the bar is such that the center of the bar coincides with the center of the ring at some point in time, then the bar will be Lorentz-contracted due to the forward component of its motion, and it will pass through the ring. The paradox occurs when the problem is considered in the rest frame of the bar. The ring is now moving downward and to the left, and will be Lorentz-contracted along its horizontal length, while the bar will not be contracted at all. How can the bar pass through the ring?

The resolution of the paradox again lies in the relativity of simultaneity (Ferraro 2007). The length of a physical object is defined as the distance between two simultaneous events occurring at each end of the body, and since simultaneity is relative, so is this length. This variability in length is just the Lorentz contraction. Similarly, a physical angle is defined as the angle formed by three simultaneous events, and this angle will also be a relative quantity. In the above paradox, although the rod and the plane of the ring are parallel in the rest frame of the ring, they are not parallel in the rest frame of the rod. The uncontracted rod passes through the Lorentz-contracted ring because the plane of the ring is rotated relative to the rod by an amount sufficient to let the rod pass through.

In mathematical terms, a Lorentz transformation can be separated into the product of a spatial rotation and a "proper" Lorentz transformation which involves no spatial rotation. The mathematical resolution of the bar and ring paradox is based on the fact that the product of two proper Lorentz transformations may produce a Lorentz transformation which is not proper, but rather includes a spatial rotation component.

See also

References

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    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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  2. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

Further reading

  • Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics (2nd ed) (Freeman, NY, 1992)
- discusses various apparent SR paradoxes and their solutions
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

  • Special Relativity Animations from John de Pillis.This inter-active animated train-and-tunnel paradox is an analog of the pole (train) and barn (tunnel) paradox.

bs:Paradoks ljestava de:Paradoxon der Längenkontraktion fr:Paradoxe du train hr:Paradoks ljestava lt:Kopėčių paradoksas hu:Létraparadoxon

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