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{{Introductory article|General relativity}}
In [[mathematics]], the '''Riemann–Siegel theta function''' is defined in terms of the [[Gamma function]] as
[[File:Cassini-science-br.jpg|thumb|right|250px|High-precision test of general relativity by the [[Cassini-Huygens|Cassini]] space probe (artist's impression): [[radio]] signals sent between the Earth and the probe (green wave) are [[Shapiro effect|delayed]] by the warping of [[space and time]] (blue lines) due to the [[Sun]]'s mass.]]
{{General relativity}}


'''General relativity''' (GR) is a [[theory]] of [[gravitation]] that was developed by [[Albert Einstein]] between 1907 and 1915. According to general relativity, the observed gravitational attraction between [[mass]]es results from their warping of [[space and time]].
:<math>\theta(t) = \arg \left(
\Gamma\left(\frac{2it+1}{4}\right)
\right)
- \frac{\log \pi}{2} t</math>


By the beginning of the 20th century, [[Newton's law of universal gravitation]] had been accepted for more than two hundred years as a valid description of the gravitational force between masses. In Newton's model, gravity is the result of an attractive force between massive objects. Although even Newton was troubled by the unknown nature of that force,<ref>- ''The Construction of Modern Science: Mechanisms and Mechanics'', by Richard S. Westfall. Cambridge University Press. 1978</ref> the basic framework was extremely successful at describing motion.
for real values of t.  Here the argument is chosen in such a way that a continuous function is obtained and <math>\theta(0)=0</math> holds, i.e., in the same way that the [[principal branch]] of the [[logarithm|log]] Gamma function is defined.


Experiments and observations show that Einstein's description of gravitation accounts for several effects that are unexplained by Newton's law, such as minute anomalies in the [[orbit]]s of [[Mercury (planet)|Mercury]] and other [[planet]]s. General relativity also predicts novel effects of gravity, such as [[gravitational wave]]s, [[gravitational lens]]ing and an effect of gravity on [[time]] known as [[gravitational time dilation]]. Many of these predictions have been confirmed by experiment, while others are the subject of ongoing research. For example, although there is indirect evidence for gravitational waves, direct evidence of their existence is still being sought by several teams of scientists in experiments such as the [[LIGO]] and [[GEO 600]] projects.
It has an [[asymptotic expansion]]


General relativity has developed into an essential tool in modern [[astrophysics]]. It provides the foundation for the current understanding of [[black hole]]s, regions of space where gravitational attraction is so strong that not even light can escape. Their strong gravity is thought to be responsible for the intense [[radiation]] emitted by certain types of astronomical objects (such as [[active galactic nucleus|active galactic nuclei]] or [[microquasar]]s). General relativity is also part of the framework of the standard [[Big Bang]] model of [[physical cosmology|cosmology]].
:<math>\theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} - \frac{t}{2} - \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots</math>


Although general relativity is not the only relativistic theory of gravity, it is the simplest such theory that is consistent with the experimental data. Nevertheless, a number of open questions remain, the most fundamental of which is how general relativity can be reconciled with the laws of [[Introduction to quantum mechanics|quantum physics]] to produce a complete and self-consistent theory of [[quantum gravity]].
which is not convergent, but whose first few terms give a good approximation for <math>t \gg 1</math>. Its Taylor-series at 0 which converges for <math>|t| < 1/2</math> is


== From special to general relativity ==
<math>\theta(t) = -\frac{t}{2} \log \pi + \sum_{k=0}^{\infty} \frac{(-1)^k \psi^{(2k)}\left(\frac{1}{4}\right) }{(2k+1)!} \left(\frac{t}{2}\right)^{2k+1}</math>
In September 1905, Albert Einstein published his theory of [[special relativity]], which reconciles [[Newton's laws of motion]] with [[electrodynamics]] (the interaction between objects with [[electric charge]]).  Special relativity introduced a new framework for all of physics by proposing new concepts of [[space]] and [[time]].  Some then-accepted physical theories were inconsistent with that framework; a key example was Newton's theory of [[gravity]], which describes the mutual attraction experienced by bodies due to their [[mass]].


Several physicists, including Einstein, searched for a theory that would reconcile Newton's law of gravity and special relativity.  Only Einstein's theory proved to be consistent with experiments and observations. To understand the theory's basic ideas, it is instructive to follow Einstein's thinking between 1907 and 1915, from his simple [[thought experiment]] involving an observer in free fall to his fully geometric theory of gravity.<ref>This development is traced e.g. in {{Harvnb|Renn|2005|loc=p. 110ff.}}, in chapters 9 through 15 of {{Harvnb|Pais|1982}}, and in {{Harvnb|Janssen|2005}}. A precis of Newtonian gravity can be found in {{Harvnb|Schutz|2003|loc=chapters 2–4}}.  It is impossible to say whether the problem of Newtonian gravity crossed Einstein's mind before 1907, but by his own admission, his first serious attempts to reconcile that theory with special relativity date to that year, cf. {{Harvnb|Pais|1982|loc=p. 178}}.</ref>
where <math>\psi^{(2k)}</math> denotes the [[Polygamma function]] of order <math>2k</math>.
The Riemann–Siegel theta function is of interest in studying the [[Riemann zeta function]], since it can rotate the Riemann zeta function such that it becomes the totally real valued [[Z function]] on the [[critical line]] <math>s = 1/2 + i t</math> .


=== Equivalence principle ===
== Curve discussion ==
{{main|Equivalence principle}}
A person in a [[free fall|free-fall]]ing elevator experiences weightlessness, and objects either float motionless or drift at constant speed.  Since everything in the elevator is falling together, no gravitational effect can be observed.  In this way, the experiences of an observer in free fall are indistinguishable from those of an observer in deep space, far from any significant source of gravity.  Such observers are the privileged ("inertial") observers Einstein described in his theory of [[special relativity]]: observers for whom [[light]] travels along straight lines at constant speed.<ref>This is described in detail in chapter 2 of {{Harvnb|Wheeler|1990}}.</ref>


Einstein hypothesized that the similar experiences of weightless observers and inertial observers in special relativity represented a fundamental property of gravity, and he made this the cornerstone of his theory of general relativity, formalized in his [[equivalence principle]]. Roughly speaking, the principle states that a person in a free-falling elevator cannot tell that he is in free fall. Every experiment in such a free-falling environment has the same results as it would for an observer at rest or moving uniformly in deep space, far from all sources of gravity.<ref>While the equivalence principle is still part of modern expositions of general relativity, there are some differences between the modern version and Einstein's original concept, cf. {{Harvnb|Norton|1985}}.</ref>
The Riemann–Siegel theta function is an odd [[real analytic function]] for real values of ''t''. It has 3 roots at 0 and <math>\pm 17.8455995405\ldots</math> and it is an increasing function for values |''t''| > 6.29, because it has exactly one minima and one maxima at <math>\pm 6.289835988\ldots</math> with absolute value <math>
3.530972829\ldots</math>. Lastly it has a unique inflection point at t=0 with <math>\theta^\prime(0)= -\frac{\ln \pi + \gamma + \pi/2 + 3 \ln 2}{2} = -2.6860917\ldots</math> where the theta function has its derivation minimum.


=== Gravity and acceleration ===
==Theta as a function of a complex variable==
[[File:Elevator gravity.svg|thumb|right|250px|Ball falling to the floor in an accelerating rocket (left) and on Earth (right)]]
We have an infinite series expression for the log Gamma function
Most effects of gravity vanish in free fall, but effects that seem the same as those of gravity can be ''produced'' by an [[Acceleration|accelerated]] frame of reference. An observer in a closed room cannot tell which of the following is true:
* Objects are falling to the floor because the room is resting on the surface of the Earth and the objects are being pulled down by gravity.
* Objects are falling to the floor because the room is aboard a rocket in space, which is accelerating at 9.81 [[metre per second squared|m/s<sup>2</sup>]] and is far from any source of gravity. The objects are being pulled towards the floor by the same "inertial force" that presses the driver of an accelerating car into the back of his seat.


Conversely, any effect observed in an accelerated reference frame should also be observed in a gravitational field of corresponding strength. This principle allowed Einstein to predict several novel effects of gravity in 1907, as explained in the [[#Physical consequences|next section]].
:<math>\log \Gamma \left(z\right) = -\gamma z -\log z  
+ \sum_{n=1}^\infty
\left(\frac{z}{n} - \log \left(1+\frac{z}{n}\right)\right),</math>


An observer in an accelerated reference frame must introduce what physicists call [[fictitious forces]] to account for the acceleration experienced by himself and objects around him. One example, the force pressing the driver of an accelerating car into his or her seat, has already been mentioned; another is the force you can feel pulling your arms up and out if you attempt to spin around like a top.  Einstein's master insight was that the constant, familiar pull of the Earth's gravitational field is fundamentally the same as these fictitious forces.<ref>E. g. {{Harvnb|Janssen|2005|loc=p. 64f}}. Einstein himself also explains this in section XX of his non-technical book Einstein 1961. Following earlier ideas by [[Ernst Mach]], Einstein also explored [[centrifugal force (fictitious)|centrifugal force]]s and their gravitational analogue, cf. {{Harvnb|Stachel|1989}}.</ref> The apparent magnitude of the fictitious forces always appears to be proportional to the [[mass]] of any object on which they act - for instance, the driver's seat exerts just enough force to accelerate the driver at the same rate as the car.  By analogy, Einstein proposed that an object in a gravitational field should feel a gravitational force proportional to its mass, as embodied in [[Newton's law of gravitation]].<ref>Einstein explained this in section XX of Einstein 1961.  He considered an object "suspended" by a rope from the ceiling of a room aboard an accelerating rocket: from inside the room it looks as if gravitation is pulling the object down with a force proportional to its mass, but from outside the rocket it looks as if the rope is simply transferring the acceleration of the rocket to the object, and must therefore exert just the "force" to do so.</ref>
where ''γ'' is [[Euler–Mascheroni constant|Euler's constant]]. Substituting <math>(2it+1)/4</math> for ''z'' and taking the imaginary part termwise gives the following series for ''θ''(''t'')


=== Physical consequences ===
:<math>\theta(t) = -\frac{\gamma + \log \pi}{2}t - \arctan 2t
In 1907, Einstein was still eight years away from completing the general theory of relativity. Nonetheless, he was able to make a number of novel, testable predictions that were based on his starting point for developing his new theory: the equivalence principle.<ref>More specifically, Einstein's calculations, which are described in chapter 11b of {{Harvnb|Pais|1982}}, use the equivalence principle, the equivalence of gravity and inertial forces, and the results of special relativity for the propagation of light and for accelerated observers (the latter by considering, at each moment, the instantaneous [[inertial frame of reference]] associated with such an accelerated observer).</ref>
+ \sum_{n=1}^\infty \left(\frac{t}{2n}
- \arctan\left(\frac{2t}{4n+1}\right)\right).</math>


[[File:Gravitational red-shifting2.png|thumb|left|200px|The gravitational redshift of a light wave as it moves upwards against a gravitational field (caused by the yellow star below)]]
For values with imaginary part between -1 and 1, the arctangent function is [[holomorphic function|holomorphic]], and it is easily seen that the series converges uniformly on compact sets in the region with imaginary part between -1/2 and 1/2, leading to a holomorphic function on this domain. It follows that the [[Z function]] is also holomorphic in this region, which is the critical strip.


The first new effect is the [[gravitational frequency shift]] of light. Consider two observers aboard an accelerating rocket-ship.  Aboard such a ship, there is a natural concept of "up" and "down": the direction in which the ship accelerates is "up", and unattached objects accelerate in the opposite direction, falling "downward". Assume that one of the observers is "higher up" than the other.  When the lower observer sends a light signal to the higher observer, the acceleration causes the light to be [[red-shift]]ed, as may be calculated from [[special relativity]]; the second observer will measure a lower [[frequency]] for the light than the first.  Conversely, light sent from the higher observer to the lower is [[blue-shift]]ed, that is, shifted towards higher frequencies.<ref>This effect can be derived directly within special relativity, either by looking at the equivalent situation of two observers in an accelerated rocket-ship or by looking at a falling elevator; in both situations, the frequency shift has an equivalent description as a [[Doppler shift]] between certain inertial frames.  For simple derivations of this, see {{Harvnb|Harrison|2002}}.</ref> Einstein argued that such frequency shifts must also be observed in a gravitational field. This is illustrated in the figure at left, which shows a light wave that is gradually red-shifted as it works its way upwards against the gravitational acceleration.  This effect has been confirmed experimentally, as described [[#Experimental tests|below]].
We may use the identities


This gravitational frequency shift corresponds to a [[gravitational time dilation]]: Since the "higher" observer measures the same light wave to have a lower frequency than the "lower" observer, time must be passing faster for the higher observer.  Thus, time runs more slowly for observers who are lower in a gravitational field.
:<math>\arg z = \frac{\log z - \log\bar z}{2i}\quad\text{and}\quad\overline{\Gamma(z)}=\Gamma(\bar z)</math>


It is important to stress that, for each observer, there are no observable changes of the flow of time for events or processes that are at rest in his or her reference frame.  Five-minute-eggs as timed by each observer's clock have the same consistency; as one year passes on each clock, each observer ages by that amount; each clock, in short, is in perfect agreement with all processes happening in its immediate vicinity.  It is only when the clocks are compared between separate observers that one can notice that time runs more slowly for the lower observer than for the higher.<ref>See chapter 12 of {{Harvnb|Mermin|2005}}.</ref>  This effect is minute, but it too has been confirmed experimentally in multiple experiments, as described [[#Experimental tests|below]].
to obtain the closed-form expression


In a similar way, Einstein predicted the [[Tests of general relativity#Deflection of light by the Sun|gravitational deflection of light]]: in a gravitational field, light is deflected downward.  Quantitatively, his results were off by a factor of two; the correct derivation requires a more complete formulation of the theory of general relativity, not just the equivalence principle.<ref>Cf. {{Harvnb|Ehlers|Rindler|1997}}; for a non-technical presentation, see {{Harvnb|Pössel|2007}}.</ref>
:<math>\theta(t) = \frac{\log\Gamma\left(\frac{2it+1}{4}\right)-\log\Gamma\left(\frac{-2it+1}{4}\right)}{2i} - \frac{\log \pi}{2} t,</math>


=== Tidal effects ===
which extends our original definition to a holomorphic function of t. Since the principal branch of log Γ has a single branch cut along the negative real axis, θ(t) in this definition inherits branch cuts along the imaginary axis above i/2 and below -i/2.
[[File:Tide fall.png|thumb|150px|Two bodies falling towards the center of the Earth accelerate towards each other as they fall.]]
The equivalence between gravitational and inertial effects does not constitute a complete theory of gravity. When it comes to explaining gravity near our own location on the Earth's surface, noting that our reference frame is not in free fall, so that [[fictitious force]]s are to be expected, provides a suitable explanation. But a freely falling reference frame on one side of the Earth cannot explain why the people on the opposite side of the Earth experience a gravitational pull in the opposite direction.


A more basic manifestation of the same effect involves two bodies that are falling side by side towards the Earth. In a reference frame that is in free fall alongside these bodies, they appear to hover weightlessly&nbsp;– but not exactly so. These bodies are not falling in precisely the same direction, but towards a single point in space: namely, the Earth's [[center of mass|center of gravity]]. Consequently, there is a component of each body's motion towards the other (see the figure). In a small environment such as a freely falling lift, this relative acceleration is minuscule, while for [[skydiver]]s on opposite sides of the Earth, the effect is large. Such differences in force are also responsible for the [[tides]] in the Earth's oceans, so the term "[[tidal effect]]" is used for this phenomenon.
{| style="text-align:center"
|+ '''Riemann–Siegel theta function in the complex plane'''
|[[Image:Riemann Siegel Theta 1.jpg|1000x140px|none]]
|[[Image:Riemann Siegel Theta 2.jpg|1000x140px|none]]
|[[Image:Riemann Siegel Theta 3.jpg|1000x140px|none]]
|-
|<math>
-1 < \Re(t) < 1
</math>
|<math>
-5 < \Re(t) < 5
</math>
|<math>
-40 < \Re(t) < 40
</math>
|}


The equivalence between inertia and gravity cannot explain tidal effects&nbsp;– it cannot explain variations in the gravitational field.<ref>These and other tidal effects are described in {{Harvnb|Wheeler|1990|pp=83–91}}.</ref> For that, a theory is needed which describes the way that matter (such as the large mass of the Earth) affects the inertial environment around it.
==Gram points==
The Riemann zeta function on the critical line can be written


=== From acceleration to geometry ===
:<math>\zeta\left(\frac{1}{2}+it\right) = e^{-i \theta(t)}Z(t),</math>
In exploring the equivalence of gravity and acceleration as well as the role of tidal forces, Einstein discovered several analogies with the [[geometry]] of [[surface]]s.  An example is the transition from an inertial reference frame (in which free particles coast along straight paths at constant speeds) to a rotating reference frame (in which extra terms corresponding to [[fictitious force]]s have to be introduced in order to explain particle motion): this is analogous to the transition from a [[Cartesian coordinate system|Cartesian]] [[coordinate system]] (in which the coordinate lines are straight lines) to a [[curvilinear coordinates|curved coordinate system]] (where coordinate lines need not be straight).
:<math>Z(t) = e^{i \theta(t)} \zeta\left(\frac{1}{2}+it\right).</math>


A deeper analogy relates tidal forces with a property of surfaces called ''[[curvature]]''. For gravitational fields, the absence or presence of tidal forces determines whether or not the influence of gravity can be eliminated by choosing a freely falling reference frame. Similarly, the absence or presence of curvature determines whether or not a surface is [[Isometry|equivalent]] to a [[Plane (mathematics)|plane]]. In the summer of 1912, inspired by these analogies, Einstein searched for a geometric formulation of gravity.<ref>Tides and their geometric interpretation are explained in chapter 5 of {{Harvnb|Wheeler|1990}}.  This part of the historical development is traced in {{Harvnb|Pais|1982|loc=section 12b}}.</ref>
If '''<math>t</math>''' is a [[real number]], then the [[Z function]] <math>Z\left(t\right)</math> returns ''real'' values.


The elementary objects of [[geometry]]&nbsp;– [[point (geometry)|points]], [[line (mathematics)|lines]], [[triangle]]s&nbsp;– are traditionally defined in three-dimensional [[space]] or on two-dimensional [[surface]]s. In 1907, the mathematician [[Hermann Minkowski]] introduced a geometric formulation of Einstein's [[special theory of relativity]] in which the geometry included not only [[space]], but also [[time]]. The basic entity of this new geometry is four-[[dimension]]al [[spacetime]]. The orbits of moving bodies are [[world line|curves in spacetime]]; the orbits of bodies moving at constant speed without changing direction correspond to straight lines.<ref>For elementary presentations of the concept of spacetime, see the first section in chapter 2 of {{Harvnb|Thorne|1994}}, and {{Harvnb|Greene|2004|loc=p. 47–61}}. More complete treatments on a fairly elementary level can be found e.g. in {{Harvnb|Mermin|2005}} and in {{Harvnb|Wheeler|1990|loc=chapters 8 and 9}}.</ref>
Hence the zeta function on the critical line will be ''real'' when
<math>\sin\left(\,\theta(t)\,\right)=0</math>. Positive real values of '''<math>t</math>''' where this occurs are called '''Gram points''', after [[Jørgen Pedersen Gram|J. P. Gram]], and can of course also be described as the points where <math>\frac{\theta(t)}{\pi}</math> is an integer.


For surfaces, the generalization from the geometry of a plane&nbsp;– a flat surface&nbsp;– to that of a general curved surface had been described in the early 19th century by [[Carl Friedrich Gauss]]. This description had in turn been generalized to higher-dimensional spaces in a mathematical formalism introduced by [[Bernhard Riemann]] in the 1850s. With the help of [[Riemannian geometry]], Einstein formulated a geometric description of gravity in which Minkowski's spacetime is replaced by distorted, curved spacetime, just as curved surfaces are a generalization of ordinary plane surfaces.<ref>See {{Harvnb|Wheeler|1990|loc=chapters 8 and 9}} for vivid illustrations of curved spacetime.</ref>
A '''Gram point''' is a solution '''<math>g_{n}</math>''' of
:<math>\theta\left(g_{n}\right) = n\pi.</math>


After he had realized the validity of this geometric analogy, it took Einstein a further three years to find the missing cornerstone of his theory: the equations describing how [[matter]] influences spacetime's curvature. Having formulated what are now known as [[Einstein's equations]] (or, more precisely, his field equations of gravity), he presented his new theory of gravity at several sessions of the [[Prussian Academy of Sciences]] in late 1915.<ref>Einstein's struggle to find the correct field equations is traced in chapters 13–15 of {{Harvnb|Pais|1982}}.</ref>
Here are the smallest non negative '''Gram points'''
{| class="wikitable" border="1"
|-
! <math>n</math>
! <math>g_{n}</math>
! <math>\theta(g_{n})</math>
|-
|style="text-align:right;"| -3
|style="text-align:right;"|  0
|style="text-align:right;"| 0
|-
|style="text-align:right;"| -2
|style="text-align:right;"|  3.4362182261...
|style="text-align:right;"| -π
|-
|style="text-align:right;"| -1
|style="text-align:right;"|  9.6669080561...
|style="text-align:right;"| -π
|-
|style="text-align:right;"| 0
| 17.8455995405...
|style="text-align:right;"| 0
|-
|style="text-align:right;"| 1
| 23.1702827012...
|style="text-align:right;"| π
|-
|style="text-align:right;"| 2
| 27.6701822178...
|style="text-align:right;"| 2π
|-
|style="text-align:right;"| 3
| 31.7179799547...
|style="text-align:right;"| 3π
|-
|style="text-align:right;"| 4
| 35.4671842971...
|style="text-align:right;"| 4π
|-
|style="text-align:right;"| 5
| 38.9992099640...
|style="text-align:right;"| 5π
|-
|style="text-align:right;"| 6
| 42.3635503920...
|style="text-align:right;"| 6π
|-
|style="text-align:right;"| 7
| 45.5930289815...
|style="text-align:right;"| 7π
|-
|style="text-align:right;"| 8
| 48.7107766217...
|style="text-align:right;"| 8π
|-
|style="text-align:right;"| 9
| 51.7338428133...
|style="text-align:right;"| 9π
|-
|style="text-align:right;"| 10
| 54.6752374468...
|style="text-align:right;"| 10π
|-
|style="text-align:right;"| 11
| 57.5451651795...
|style="text-align:right;"| 11π
|-
|style="text-align:right;"| 12
| 60.3518119691...
|style="text-align:right;"| 12π
|-
|style="text-align:right;"| 13
| 63.1018679824...
|style="text-align:right;"| 13π
|-
|style="text-align:right;"| 14
| 65.8008876380...
|style="text-align:right;"| 14π
|-
|style="text-align:right;"| 15
| 68.4535449175...
|style="text-align:right;"| 15π
|}


== Geometry and gravitation ==
The choice of the index ''n'' is a bit crude. It is historically chosen in such a way that the index is 0 at the first value which is larger than the smallest positive zero (at imaginary part 14.13472515 ...) of the Riemann zeta function on the critical line. Notice, this <math>\theta</math>-function oscillates for absolute-small real arguments and therefore is not uniquely invertible in the interval [-24,24]! Thus the [[Odd function|odd]] theta-function has its symmetric Gram point with value 0 at index -3.
Paraphrasing [[John Archibald Wheeler|John Wheeler]], Einstein's geometric theory of gravity can be summarized thus: '''spacetime tells matter how to move; matter tells spacetime how to curve'''.<ref>E.g. p. xi in {{Harvnb|Wheeler|1990}}.</ref> What this means is addressed in the following three sections, which explore the motion of so-called test particles, examine which properties of matter serve as a source for gravity, and, finally, introduce Einstein's equations, which relate these matter properties to the curvature of spacetime.
Gram points are useful when computing the zeros of <math>Z\left(t\right)</math>. At a Gram point '''<math>g_{n}</math>''',


=== Probing the gravitational field ===
:<math>\zeta\left(\frac{1}{2}+ig_n\right) = \cos(\theta(g_n))Z(g_n) = (-1)^n Z(g_n),</math>
[[File:Earth geo.gif|thumb|right|250px|Converging geodesics: two lines of longitude (green) that start out in parallel at the equator (red) but converge to meet at the pole]]


In order to map a body's gravitational influence, it is useful to think about what physicists call probe or [[test particle]]s: particles that are influenced by gravity, but are so small and light that we can [[negligible|neglect]] their own gravitational effect. In the absence of gravity and other external forces, a test particle moves along a straight line at a constant speed. In the language of [[spacetime]], this is equivalent to saying that such test particles move along straight [[world lines]] in spacetime. In the presence of gravity, spacetime is [[non-Euclidean]], or [[curvature|curved]], and in curved spacetime straight world lines may not exist. Instead, test particles move along lines called [[geodesic]]s, which are "as straight as possible".
and if this is ''positive'' at ''two'' successive Gram points, <math>Z\left(t\right)</math> must have a zero in the interval.


A simple analogy is the following: In [[geodesy]], the science of measuring Earth's size and shape, a geodesic (from Greek "geo", Earth, and "daiein", to divide) is the shortest route between two points on the Earth's surface. Approximately, such a route is a [[Circle segment|segment]] of a [[great circle]], such as a [[line of longitude]] or the [[equator]]. These paths are certainly not straight, simply because they must follow the curvature of the Earth's surface. But they are as straight as is possible subject to this constraint.
According to '''Gram’s law''', the [[real part]] is ''usually'' positive while the [[imaginary part]] alternates with the '''gram points''', between ''positive'' and ''negative'' values at somewhat regular intervals.


The properties of geodesics differ from those of straight lines. For example, on a plane, parallel lines never meet, but this is not so for geodesics on the surface of the Earth: for example, lines of longitude are parallel at the equator, but intersect at the poles. Analogously, the world lines of test particles in free fall are [[geodesic (general relativity)|spacetime geodesics]], the straightest possible lines in spacetime.  But still there are crucial differences between them and the truly straight lines that can be traced out in the gravity-free spacetime of special relativity.  In special relativity, parallel geodesics remain parallel. In a gravitational field with tidal effects, this will not, in general, be the case. If, for example, two bodies are initially at rest relative to each other, but are then dropped in the Earth's gravitational field, they will move towards each other as they fall towards the Earth's center.<ref>A thorough, yet accessible account of basic differential geometry and its application in general relativity can be found in {{Harvnb|Geroch|1978}}.</ref>
:<math>(-1)^n \, Z\left(g_{n}\right) > 0</math>


Compared with planets and other astronomical bodies, the objects of everyday life (people, cars, houses, even mountains) have little mass. Where such objects are concerned, the laws governing the behavior of test particles are sufficient to describe what happens. Notably, in order to deflect a test particle from its geodesic path, an external force must be applied. A person sitting on a chair is trying to follow a geodesic, that is, to [[free fall|fall freely]] towards the center of the Earth. But the chair applies an external upwards force preventing the person from falling. In this way, general relativity explains the daily experience of gravity on the surface of the Earth ''not'' as the downwards pull of a gravitational force, but as the upwards push of external forces. These forces deflect all bodies resting on the Earth's surface from the geodesics they would otherwise follow.<ref>See chapter 10 of {{Harvnb|Wheeler|1990}}.</ref> For matter objects whose own gravitational influence cannot be neglected, the laws of motion are somewhat more complicated than for test particles, although it remains true that spacetime tells matter how to move.<ref>In fact, when starting from the complete theory, Einstein's equation can be used to derive these more complicated laws of motion for matter as a consequence of geometry, but deriving from this the motion of idealized test particles is a highly non-trivial task, cf. {{Harvnb|Poisson|2004}}.</ref>
The number of roots, <math>N\left(T\right)</math>, in the strip from 0 to ''T'', can be found by
:<math>N\left(T\right) = \frac{\theta(T)}{\pi} + 1+S(T),</math>
where <math>S(T)</math> is an error term which grows asymptotically like <math>\log T</math>.


=== Sources of gravity ===
Only if <math>g_{n}</math> '''would obey Gram’s law''', then finding the number of roots in the strip simply becomes
In [[law of universal gravitation|Newton's description of gravity]], the gravitational force is caused by matter. More  precisely, it is caused by a specific property of material objects: their [[mass]]. In Einstein's theory and related [[theories of gravitation]], curvature at every point in spacetime is also caused by whatever matter is present. Here, too, mass is a key property in determining the gravitational influence of matter. But in a relativistic theory of gravity, mass cannot be the only source of gravity. Relativity links mass with energy, and energy with momentum.


The equivalence between mass and [[energy]], as expressed by the formula [[mass-energy equivalence|''E''&nbsp;=&nbsp;''mc''<sup>2</sup>]], is perhaps the most famous consequence of special relativity. In relativity, mass and energy are two different ways of describing one physical quantity. If a physical system has energy, it also has the corresponding mass, and vice versa. In particular, all properties of a body that are associated with energy, such as its [[temperature]] or the [[binding energy]] of systems such as [[Atomic nucleus|nuclei]] or [[molecules]], contribute to that body's mass, and hence act as sources of gravity.<ref>A simple explanation of mass-energy-equivalence can be found in sections 3.8 and 3.9 of {{Harvnb|Giulini|2005}}.</ref>
:<math>N\left(g_{n}\right) = n + 1.</math>


In special relativity, energy is closely connected to [[momentum]]. Just as space and time are, in that theory, different aspects of a more comprehensive entity called spacetime, energy and momentum are merely different aspects of a unified, four-dimensional quantity that physicists call [[four-momentum]].  In consequence, if energy is a source of gravity, momentum must be a source as well. The same is true for quantities that are directly related to energy and momentum, namely internal [[pressure]] and [[Tension (physics)|tension]]. Taken together, in general relativity it is mass, energy, momentum, pressure and tension that serve as sources of gravity: they are how matter tells spacetime how to curve. In the theory's mathematical formulation, all these quantities are but aspects of a more general physical quantity called the [[Stress-energy tensor|energy-momentum tensor]].<ref>See chapter 6 of {{Harvnb|Wheeler|1990}}.</ref>
Today we know, that in the long run, '''Gram's law''' fails for about 1/4 of all Gram-intervals to contain exactly 1 zero of the Riemann zeta-function. Gram was afraid that it may fail for larger indices (the first miss is at index 126 before the 127th zero) and thus claimed this only for not too high indices. Later Hutchinson coined the phrase ''Gram's law'' for the (false) statement that all zeroes on the critical line would be separated by Gram points.


=== Einstein's equations ===
==See also==
[[Einstein's equations]] are the centerpiece of general relativity. They provide a precise formulation of the relationship between spacetime geometry and the properties of matter, using the language of mathematics. More concretely, they are formulated using the concepts of [[Riemannian geometry]], in which the geometric properties of a space (or a spacetime) are described by a quantity called a [[Riemannian metric|metric]]. The metric encodes the information needed to compute the fundamental geometric notions of distance and angle in a curved space (or spacetime).
*[[Z function]]
 
[[File:Metric globe.png|thumb|right|250px|Distances corresponding to 30 degrees difference in longitude, at different latitudes]]
A spherical surface like that of the Earth provides a simple example. The location of any point on the surface can be described by two coordinates: the geographic [[latitude]] and [[longitude]]. Unlike the Cartesian coordinates of the plane, coordinate differences are not the same as distances on the surface, as shown in the diagram on the right: for someone at the equator, moving 30 degrees of longitude westward (magenta line) corresponds to a distance of roughly {{convert|3300|km|mi|sp=us}}. On the other hand, someone at a latitude of 55 degrees, moving 30 degrees of longitude westward (blue line) covers a distance of merely {{convert|1900|km|mi|sp=us}}. Coordinates therefore do not provide enough information to describe the geometry of a spherical surface, or indeed the geometry of any more complicated space or spacetime. That information is precisely what is encoded in the metric, which is a function defined at each point of the surface (or space, or spacetime) and relates coordinate differences to differences in distance. All other quantities that are of interest in geometry, such as the length of any given curve, or the angle at which two curves meet, can be computed from this metric function.<ref>For a more detailed definition of the metric, but one that is more informal than a textbook presentation, see chapter 14.4 of {{Harvnb|Penrose|2004}}.</ref>
 
The metric function and its rate of change from point to point can be used to define a geometrical quantity called the [[Riemann curvature tensor]], which describes exactly how the space (or spacetime) is curved at each point. In general relativity, the metric and the Riemann curvature tensor are quantities defined at each point in spacetime. As has already been mentioned, the matter content of the spacetime defines another quantity, the [[Stress-energy tensor|Energy-momentum tensor]] '''T''', and the principle that "spacetime tells matter how to move, and matter tells spacetime how to curve" means that these quantities must be related to each other. Einstein formulated this relation by using the Riemann curvature tensor and the metric to define another geometrical quantity '''G''', now called the [[Einstein tensor]], which describes some aspects of the way spacetime is curved. ''Einstein's equation'' then states that
 
: <math>\mathbf{G}=\frac{8\pi G}{c^4}\mathbf{T},</math>
 
i.e., up to a constant multiple, the quantity '''G''' (which measures curvature) is equated with the quantity '''T''' (which measures matter content). The constants involved in this equation reflect the different theories that went into its making: ''G'' is the [[gravitational constant]] that is already present in Newtonian gravity; ''c'' is the [[speed of light]], the key constant in special relativity; and [[pi|&pi;]] is one of the basic constants of geometry.
 
This equation is often referred to in the plural as ''Einstein's equations'', since the quantities '''G''' and '''T''' are each determined by several functions of the coordinates of spacetime, and the equations equate each of these component functions.<ref>The geometrical meaning of Einstein's equations is explored in chapters 7 and 8 of {{Harvnb|Wheeler|1990}}; cf. box 2.6 in {{Harvnb|Thorne|1994}}. An introduction using only very simple mathematics is given in chapter 19 of {{Harvnb|Schutz|2003}}.</ref>  [[Exact solutions of Einstein's field equations|A solution of these equations]] describes a particular geometry of [[space and time]]; for example, the [[Schwarzschild metric|Schwarzschild solution]] describes the geometry around a spherical, non-rotating mass such as a [[star]] or a [[black hole]], whereas the [[Kerr metric|Kerr solution]] describes a rotating black hole.  Still other solutions can describe a [[gravitational wave]] or, in the case of the [[Friedmann–Lemaître–Robertson–Walker metric|Friedmann–Lemaître–Robertson–Walker solution]], an expanding universe.  The simplest solution is the uncurved [[Minkowski spacetime]], the spacetime described by special relativity.<ref>The most important solutions are listed in every [[General relativity#Further reading|textbook on general relativity]]; for a (technical) summary of our current understanding, see {{Harvnb|Friedrich|2005}}.</ref>
 
==Experimental tests==
No scientific theory is [[apodicticity|apodictically true]]; each is a model that must be checked by experiment. [[Newton's law of gravity]] was accepted because it accounted for the motion of planets and moons in the solar system with exquisite accuracy.  As the precision of experimental measurements gradually improved, some discrepancies with Newton's predictions were observed, and these were accounted for in the general theory of relativity.  Similarly, the predictions of general relativity must also be checked with experiment, and Einstein himself devised three tests now known as the classical tests of the theory:
[[File:Newtonianvseinsteinianorbits.gif|thumb|right|200px|Newtonian (red) vs. Einsteinian orbit (blue) of a single planet orbiting a spherical star. Click on the image for animation.]]
*Newtonian gravity predicts that the [[orbit]] which a single [[planet]] traces around a perfectly spherical [[star]] should be an [[ellipse]]. Einstein's theory predicts a more complicated curve: the planet behaves as if it were travelling around an ellipse, but at the same time, the ellipse as a whole is rotating slowly around the star. In the diagram on the right, the ellipse predicted by Newtonian gravity is shown in red, and part of the orbit predicted by Einstein in blue. For a planet orbiting the [[Sun]], this deviation from Newton's orbits is known as the [[Tests of general relativity#Perihelion precession of Mercury|anomalous perihelion shift]].  The first measurement of this effect, for the planet [[Mercury (planet)|Mercury]], dates back to 1859. The most accurate results for Mercury and for other planets to date are based on measurements which were undertaken between 1966 and 1990, using [[radio telescope]]s.<ref>More precisely, these are [[VLBI]] measurements of planetary positions; see chapter 5 of {{Harvnb|Will|1993}} and section 3.5 of {{Harvnb|Will|2006}}.</ref>  General relativity predicts the correct anomalous perihelion shift for all planets where this can be measured accurately ([[Mercury (planet)|Mercury]], [[Venus]] and the [[Earth]]).
*According to general relativity, light does not travel along straight lines when it propagates in a gravitational field. Instead, it is [[Tests of general relativity#Deflection of light by the Sun|deflected]] in the presence of massive bodies. In particular, starlight is deflected as it passes near the [[Sun]], leading to apparent shifts of up 1.75 [[Minute of arc|arc second]]s in the stars' positions in the night sky (an arc second is equal to 1/3600 of a [[Degree (angle)|degree]]). In the framework of Newtonian gravity, a heuristic argument can be made that leads to light deflection by half that amount. The different predictions can be tested by observing stars that are close to the Sun during a [[solar eclipse]]. In this way, a British expedition to West Africa in 1919, directed by [[Arthur Eddington]], confirmed that Einstein's prediction was correct, and the Newtonian predictions wrong, via observation of the [[Solar eclipse of May 29, 1919|May 1919 eclipse]]. Eddington's results were not very accurate; subsequent observations of the deflection of the light of distant [[quasar]]s by the Sun, which utilize highly accurate techniques of [[radio astronomy]], have confirmed Eddington's results with significantly better precision (the first such measurements date from 1967, the most recent comprehensive analysis from 2004).<ref>For the historical measurements, see {{Harvnb|Hartl|2005}}, {{Harvnb|Kennefick|2005}}, and {{Harvnb|Kennefick|2007}}; Soldner's original derivation in the framework of Newton's theory is {{Harvnb|Soldner|1804}}. For the most precise measurements to date, see {{Harvnb|Bertotti|2005}}.</ref>
*[[Gravitational redshift]] was first measured in a laboratory setting in 1959 by [[Pound-Rebka experiment|Pound and Rebka]]. It is also seen in astrophysical measurements, notably for light escaping the [[White Dwarf]] [[Sirius B]]. The related [[gravitational time dilation]] effect has been measured by transporting [[atomic clock]]s to altitudes of between tens and tens of thousands of kilometers (first by [[Hafele-Keating experiment|Hafele and Keating]] in 1971; most accurately to date by [[Gravity Probe A]] launched in 1976).<ref>See {{Harvnb|Kennefick|2005}} and chapter 3 of {{Harvnb|Will|1993}}. For the Sirius B measurements, see {{Harvnb|Trimble|Barstow|2007}}.</ref>
Of these tests, only the perihelion advance of Mercury was known prior to Einstein's final publication of general relativity in 1916.  The subsequent experimental confirmation of his other predictions, especially the first measurements of the deflection of light by the sun in 1919, catapulted Einstein to international stardom.<ref>{{Harvnb|Pais|1982}}, Mercury on pp. 253–254, Einstein's rise to fame in sections 16b and 16c.</ref>  These three experimental tests justified adopting general relativity over Newton's theory and, incidentally, over a number of [[alternatives to general relativity]] that had been proposed.
 
[[File:Gravity Probe B.jpg|thumb|left|200px|[[Gravity Probe B]] with its solar panels folded]]
Further tests of general relativity include precision measurements of the [[Shapiro effect]] or gravitational time delay for light, most recently in 2002 by the [[Cassini-Huygens|Cassini]] space probe. One set of tests focuses on effects predicted by general relativity for the behavior of [[gyroscopes]] travelling through space. One of these effects, [[geodetic effect|geodetic precession]], has been tested with the [[Lunar Laser Ranging Experiment]] (high precision measurements of the orbit of the [[Moon]]). Another, which is related to rotating masses, is called [[frame-dragging]]. The geodetic and frame-dragging effects were both tested by the [[Gravity Probe B]] satellite experiment launched in 2004, with results confirming relativity to within 0.5% and 15%, respectively, as of December 2008.<ref>
{{cite web
|author=Everitt, C.W.F.; Parkinson, B.W.
|url=http://einstein.stanford.edu/content/final_report/GPB_Final_NASA_Report-020509-web.pdf
|title=Gravity Probe B Science Results—NASA Final Report
|format=PDF
|year=2009
|accessdate=2009-05-02
}}</ref>
 
By cosmic standards, gravity throughout the solar system is weak. Since the differences between the predictions of Einstein's and Newton's theories are most pronounced when gravity is strong, physicists have long been interested in testing various relativistic effects in a setting with comparatively strong gravitational fields. This has become possible thanks to precision observations of [[binary pulsar]]s. In such a star system, two highly compact [[neutron star]]s orbit each other. At least one of them is a [[pulsar]]&nbsp;– an astronomical object that emits a tight beam of radiowaves. These beams strike the Earth at very regular intervals, similarly to the way that the rotating beam of a lighthouse means that an observer sees the lighthouse blink,  and can be observed as a highly regular series of pulses. General relativity predicts specific deviations from the regularity of these radio pulses. For instance, at times when the radio waves pass close to the other neutron star, they should be deflected by the star's gravitational field. The observed pulse patterns are impressively close to those predicted by general relativity.<ref>{{Harvnb|Kramer|2004}}.</ref>
 
One particular set of observations is related to eminently useful practical applications, namely to [[satellite navigation system]]s such as the [[Global Positioning System]] that are used both for precise [[navigation|positioning]] and [[timekeeping]]. Such systems rely on two sets of [[atomic clocks]]: clocks aboard satellites orbiting the Earth, and reference clocks stationed on the Earth's surface. General relativity predicts that these two sets of clocks should tick at slightly different rates, due to their different motions (an effect already predicted by special relativity) and their different positions within the Earth's gravitational field. In order to ensure the system's accuracy, the satellite clocks are either slowed down by a relativistic factor, or that same factor is made part of the evaluation algorithm. In turn, tests of the system's accuracy (especially the very thorough measurements that are part of the definition of [[UTC|universal coordinated time]]) are testament to the validity of the relativistic predictions.<ref>An accessible account of relativistic effects in the global positioning system can be found in {{Harvnb|Ashby|2002}}; details are given in {{Harvnb|Ashby|2003}}.</ref>
 
A number of other tests have probed the validity of various versions of the [[equivalence principle]]; strictly speaking, all measurements of gravitational time dilation are tests of the [[weak equivalence principle|weak]] version of that principle, not of general relativity itself. So far, general relativity has passed all observational tests.<ref>An accessible introduction to tests of general relativity is {{Harvnb|Will|1993}}; a more technical, up-to-date account is {{Harvnb|Will|2006}}.</ref>
 
==Astrophysical applications==
Models based on general relativity play an important role in [[astrophysics]], and the success of these models is further testament to the theory's validity.
 
===Gravitational lensing===
[[File:Einstein cross.jpg|thumb|220px|[[Einstein cross]]: four images of the same astronomical object, produced by a [[gravitational lens]]]]
 
Since light is deflected in a gravitational field, it is possible for the light of a distant object to reach an observer along two or more paths. For instance, light of a very distant object such as a [[quasar]] can pass along one side of a massive [[galaxy]] and be deflected slightly so as to reach an observer on Earth, while light
passing along the opposite side of that same galaxy is deflected as well, reaching the same observer from a slightly different direction. As a result, that particular observer will see one astronomical object in two different places in the night sky.  This kind of focussing is well-known when it comes to [[optical lens]]es, and hence the corresponding gravitational effect is called [[gravitational lensing]].<ref>The geometry of such situations is explored in chapter 23 of {{Harvnb|Schutz|2003}}.</ref>
 
[[Observational astronomy]] uses lensing effects as an important tool to infer properties of the lensing object. Even in cases where that object is not directly visible, the shape of a lensed image provides information about the [[mass]] distribution responsible for the light deflection. In particular, gravitational lensing provides one way to measure the distribution of [[dark matter]], which does not give off light and can be observed only by its gravitational effects. One particularly interesting application are large-scale observations, where the lensing masses are spread out over a significant fraction of the observable universe, and can be used to obtain information about the large-scale properties and evolution of our cosmos.<ref>Introductions to gravitational lensing and its applications can be found on the webpages  {{Harvnb|Newbury|1997}} and {{Harvnb|Lochner|2007}}.</ref>
 
===Gravitational waves===
[[Gravitational wave]]s, a direct consequence of Einstein's theory, are distortions of geometry that propagate at the speed of light, and can be thought of as ripples in spacetime. They should not be confused with the [[gravity wave]]s of [[fluid dynamics]], which are a different concept.
 
Indirectly, the effect of gravitational waves has been detected in observations of specific binary stars. Such pairs of stars [[orbit]] each other and, as they do so, gradually lose energy by emitting gravitational waves. For ordinary stars like our sun, this energy loss would be too small to be detectable, but this energy loss was observed in 1974 in a [[binary pulsar]] called [[PSR1913+16]]. In such a system, one of the orbiting stars is a pulsar. This has two consequences: a pulsar is an extremely dense object known as a [[neutron star]], for which gravitational wave emission is much stronger than for ordinary stars. Also, a pulsar emits a narrow beam of [[electromagnetic radiation]] from its magnetic poles. As the pulsar rotates, its beam sweeps over the Earth, where it is seen as a regular series of radio pulses, just as a ship at sea observes regular flashes of light from the rotating light in a lighthouse. This regular pattern of radio pulses functions as a highly accurate "clock". It can be used to time the double star's orbital period, and it reacts sensitively to distortions of space-time in its immediate neighborhood.
 
The discoverers of PSR1913+16, [[Russell Alan Hulse|Russell Hulse]] and [[Joseph Hooton Taylor Jr.|Joseph Taylor]], were awarded the [[Nobel Prize in Physics]] in 1993. Since then, several other binary pulsars have been found. The most useful are those in which both stars are pulsars, since they provide the most accurate tests of general relativity.<ref>{{Harvnb|Schutz|2003|loc=pp. 317–321}}; {{Harvnb|Bartusiak|2000|loc=pp. 70–86}}.</ref>
 
Currently, one major goal of research in relativity is the direct detection of gravitational waves. To this end, a number of land-based [[gravitational wave detector]]s are in operation, and a mission to launch a space-based detector, [[Laser Interferometer Space Antenna|LISA]], is currently under development, with a precursor mission ([[LISA Pathfinder]]) due for launch in June 2013. If gravitational waves are detected, they could be used to obtain information about compact objects such as [[neutron star]]s and [[black holes]], and also to probe the state of the early [[universe]] fractions of a [[second]] after the [[Big Bang]].<ref>The ongoing search for gravitational waves is described vividly in {{Harvnb|Bartusiak|2000}} and in {{Harvnb|Blair|McNamara|1997}}.</ref>
 
===Black holes===
[[File:M87 jet.jpg|thumb|right|200px|Black hole-powered jet emanating from the central region of the galaxy [[Messier 87|M87]]]]
When mass is concentrated into a sufficiently [[hoop conjecture|compact]] region of space, general relativity predicts the formation of a [[black hole]]&nbsp;– a region of space with a gravitational attraction so strong that not even light can escape. Certain types of black holes are thought to be the final state in the [[stellar evolution|evolution]] of massive [[star]]s. On the other hand, [[supermassive black hole]]s with the mass of [[million]]s or [[1000000000 (number)|billion]]s of [[Sun]]s are assumed to reside in the cores of most [[galaxy|galaxies]], and they play a key role in current models of how galaxies have formed over the past billions of years.<ref>For an overview of the history of black hole physics from its beginnings in the early 20th century to modern times, see the very readable account by {{Harvnb|Thorne|1994}}.  For an up-to-date account of the role of black holes in structure formation, see {{Harvnb|Springel|White|Jenkins|Frenk|2005}}; a brief summary can be found in the related article {{Harvnb|Gnedin|2005}}.</ref>
 
Matter falling onto a compact object is one of the most efficient mechanisms for releasing [[energy]] in the form of [[radiation]], and matter falling onto black holes is thought to be responsible for some of the brightest astronomical phenomena imaginable. Notable examples of great interest to astronomers are [[quasars]] and other types of [[active galactic nucleus|active galactic nuclei]]. Under the right conditions, falling matter accumulating around a black hole can lead to the formation of [[relativistic jet|jet]]s, in which focused beams of matter are flung away into space at speeds near [[lightspeed|that of light]].<ref>See chapter 8 of {{Harvnb|Sparke|Gallagher|2007}} and {{Harvnb|Disney|1998}}.  A treatment that is more thorough, yet involves only comparatively little mathematics can be found in {{Harvnb|Robson|1996}}.</ref>
 
There are several properties that make black holes most promising sources of gravitational waves. One reason is that black holes are the most compact objects that can orbit each other as part of a binary system; as a result, the gravitational waves emitted by such a system are especially strong. Another reason follows from what are called [[no hair theorem|black hole uniqueness theorems]]: over time, black holes retain only a minimal set of distinguishing features (since different hair styles are a crucial part of what gives different people their different appearances, these theorems have become known as "no hair" theorems). For instance, in the long term, the collapse of a hypothetical matter cube will not result in a cube-shaped black hole. Instead, the resulting black hole will be indistinguishable from a black hole formed by the collapse of a spherical mass, but with one important difference: in its transition to a spherical shape, the black hole formed by the collapse of a cube will emit gravitational waves.<ref>An elementary introduction to the black hole uniqueness theorems can be found in {{Harvnb|Chrusciel|2006}} and in {{Harvnb|Thorne|1994|loc=pp. 272–286}}.</ref>
 
===Cosmology===
[[File:WMAP image of the CMB anisotropy.jpg|thumb|250px|Image of [[radiation]] emitted no more than a few hundred thousand years after the big bang, captured with the satellite telescope [[WMAP]]]]
One of the most important aspects of general relativity is that it can be applied to the [[universe]] as a whole. A key point is that, on large scales, our universe appears to be constructed along very simple lines: All current observations suggest that, on average, the structure of the cosmos should be approximately the same, regardless of an observer's location or direction of observation: the universe is approximately [[Homogeneity (physics)|homogeneous]] and [[isotropic]]. Such comparatively simple universes can be described by simple solutions of Einstein's equations. The current [[physical cosmology|cosmological models]] of the universe are obtained by combining these simple solutions to general relativity with theories describing the properties of the universe's [[matter]] content, namely [[thermodynamics]], [[nuclear physics|nuclear-]] and [[particle physics]]. According to these models, our present universe emerged from an extremely dense high-temperature state (the [[Big Bang]])
roughly 14 [[1000000000 (number)|billion]] [[year]]s ago, and has been [[cosmic expansion|expanding]] ever since.<ref>Detailed information can be found in Ned Wright's Cosmology Tutorial and FAQ, {{Harvnb|Wright|2007}}; a very readable introduction is {{Harvnb|Hogan|1999}}. Using undergraduate mathematics but avoiding the advanced mathematical tools of general relativity, {{Harvnb|Berry|1989}} provides a more thorough presentation.</ref>
 
Einstein's equations can be generalized by adding a term called the [[cosmological constant]].  When this term is present, [[vacuum|empty space]] itself acts as a source of attractive or, unusually, repulsive gravity. Einstein originally introduced this term in his pioneering 1917 paper on cosmology, with a very specific motivation: contemporary cosmological thought held the universe to be static, and the additional term was required for constructing static model universes within the framework of general relativity. When it became apparent that the universe is not static, but expanding, Einstein was quick to discard this additional term; prematurely, as we know today: From about 1998 on, a steadily accumulating body of astronomical evidence has shown that the expansion of the universe is [[accelerate|accelerating]] in a way that suggests the presence of a cosmological constant or, equivalently, of a [[dark energy]] with specific properties that pervades all of space.<ref>Einstein's original paper is {{Harvnb|Einstein|1917}}; good descriptions of more modern developments can be found in {{Harvnb|Cowen|2001}} and {{Harvnb|Caldwell|2004}}.</ref>
 
==Modern research: general relativity and beyond==
General relativity is very successful in providing a framework for accurate models which describe an impressive array of physical phenomena. On the other hand, there are many interesting open questions, and in particular, the theory as a whole is almost certainly incomplete.<ref>Cf. {{Harvnb|Maddox|1998|loc=pp. 52–59 and 98–122}}; {{Harvnb|Penrose|2004|loc=section 34.1 and chapter 30}}.</ref>
 
In contrast to all other modern theories of [[fundamental interaction]]s, general relativity is a [[classical physics|classical]] theory: it does not include the effects of [[Introduction to quantum mechanics|quantum physics]]. The quest for a quantum version of general relativity addresses one of the most fundamental [[List of unsolved problems in physics|open questions]] in physics. While there are promising candidates for such a theory of [[quantum gravity]], notably [[string theory]] and [[loop quantum gravity]], there is at present no consistent and complete theory. It has long been hoped that a theory of quantum gravity would also eliminate another problematic feature of general relativity: the presence of [[spacetime singularity|spacetime singularities]]. These singularities are boundaries ("sharp edges") of spacetime at which geometry becomes ill-defined, with the consequence that general relativity itself loses its predictive power. Furthermore, there are so-called [[Penrose-Hawking singularity theorems|singularity theorems]] which predict that such singularities ''must'' exist within the universe if the laws of general relativity were to hold without any quantum modifications. The best-known examples are the singularities associated with the model universes that describe black holes and the [[Big bang|beginning of the universe]].<ref>With a focus on string theory, the search for quantum gravity is described in {{Harvnb|Greene|1999}}; for an account from the point of view of loop quantum gravity, see {{Harvnb|Smolin|2001}}.</ref>
 
Other attempts to modify general relativity have been made in the context of [[cosmology]]. In the modern cosmological models, most energy in the universe is in forms that have never been detected directly, namely [[dark energy]] and [[dark matter]]. There have been several controversial proposals to obviate the need for these enigmatic forms of matter and energy, by modifying the laws governing gravity and the dynamics of [[cosmic expansion]], for example [[modified Newtonian dynamics]].<ref>For dark matter, see {{Harvnb|Milgrom|2002}}; for dark energy, {{Harvnb|Caldwell|2004}}</ref>
 
Beyond the challenges of quantum effects and cosmology, research on general relativity is rich with possibilities for further exploration:  mathematical relativists explore the nature of singularities and the fundamental properties of Einstein's equations,<ref>See {{Harvnb|Friedrich|2005}}.</ref> ever more comprehensive computer simulations of specific spacetimes (such as those describing merging black holes) are run,<ref>A review of the various problems and the techniques being developed to overcome them, see {{Harvnb|Lehner|2002}}.</ref> and the race for the first direct detection of gravitational waves continues apace.<ref>See {{Harvnb|Bartusiak|2000}} for an account up to that year; up-to-date news can be found on the websites of major detector collaborations such as [http://geo600.aei.mpg.de GEO 600] and [http://www.ligo.caltech.edu/ LIGO].</ref>
More than ninety years after the theory was first published, research is more active than ever.<ref>A good starting point for a snapshot of present-day research in relativity is the electronic review journal [http://relativity.livingreviews.org Living Reviews in Relativity].</ref>
 
== See also ==
{{portal|Physics|Astronomy}}
* [[Introduction to special relativity]]
* [[History of general relativity]]
* [[Tests of general relativity]]
* [[Numerical relativity]]
 
==Notes==
{{reflist|colwidth=30em}}


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== External links ==
{{Wikibooks|General relativity}}
{{commons category|General relativity}}
''Additional resources, including more advanced material, can be found in [[General relativity resources]].''
* [http://www.einstein-online.info/en Einstein Online]. Website featuring articles on a variety of aspects of relativistic physics for a general audience, hosted by the [[Max Planck Institute for Gravitational Physics]]
* [http://archive.ncsa.uiuc.edu/Cyberia/NumRel/NumRelHome.html NCSA Spacetime Wrinkles]. Website produced by the [[numerical relativity]] group at the [[National Center for Supercomputing Applications]], featuring an elementary introduction to general relativity, [[black hole]]s and [[gravitational wave]]s
 
{{featured article}}
 
[[Category:Gravitation]]
[[Category:General relativity| ]]
[[Category:Theories of gravitation]]


{{Link FA|vi}}
==External links==
*{{MathWorld|title=Riemann-Siegel Functions|urlname=Riemann-SiegelFunctions}}
*[http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/RiemannSiegelTheta/ Wolfram Research – Riemann-Siegel Theta function] (includes function plotting and evaluation)


[[ar:مقدمة النسبية العامة]]
{{DEFAULTSORT:Riemann-Siegel theta function}}
[[bn:সাধারণ আপেক্ষিকতার ভূমিকা]]
[[Category:Zeta and L-functions]]
[[es:Introducción a la relatividad general]]
[[fr:Introduction à la relativité générale]]
[[ms:Pengenalan kerelatifan am]]
[[vi:Giới thiệu thuyết tương đối rộng]]
[[zh:廣義相對論入門]]

Revision as of 16:53, 13 August 2014

In mathematics, the Riemann–Siegel theta function is defined in terms of the Gamma function as

for real values of t. Here the argument is chosen in such a way that a continuous function is obtained and holds, i.e., in the same way that the principal branch of the log Gamma function is defined.

It has an asymptotic expansion

which is not convergent, but whose first few terms give a good approximation for . Its Taylor-series at 0 which converges for is

where denotes the Polygamma function of order . The Riemann–Siegel theta function is of interest in studying the Riemann zeta function, since it can rotate the Riemann zeta function such that it becomes the totally real valued Z function on the critical line .

Curve discussion

The Riemann–Siegel theta function is an odd real analytic function for real values of t. It has 3 roots at 0 and and it is an increasing function for values |t| > 6.29, because it has exactly one minima and one maxima at with absolute value . Lastly it has a unique inflection point at t=0 with where the theta function has its derivation minimum.

Theta as a function of a complex variable

We have an infinite series expression for the log Gamma function

where γ is Euler's constant. Substituting for z and taking the imaginary part termwise gives the following series for θ(t)

For values with imaginary part between -1 and 1, the arctangent function is holomorphic, and it is easily seen that the series converges uniformly on compact sets in the region with imaginary part between -1/2 and 1/2, leading to a holomorphic function on this domain. It follows that the Z function is also holomorphic in this region, which is the critical strip.

We may use the identities

to obtain the closed-form expression

which extends our original definition to a holomorphic function of t. Since the principal branch of log Γ has a single branch cut along the negative real axis, θ(t) in this definition inherits branch cuts along the imaginary axis above i/2 and below -i/2.

Riemann–Siegel theta function in the complex plane

Gram points

The Riemann zeta function on the critical line can be written

If is a real number, then the Z function returns real values.

Hence the zeta function on the critical line will be real when . Positive real values of where this occurs are called Gram points, after J. P. Gram, and can of course also be described as the points where is an integer.

A Gram point is a solution of

Here are the smallest non negative Gram points

-3 0 0
-2 3.4362182261... -π
-1 9.6669080561... -π
0 17.8455995405... 0
1 23.1702827012... π
2 27.6701822178... 2π
3 31.7179799547... 3π
4 35.4671842971... 4π
5 38.9992099640... 5π
6 42.3635503920... 6π
7 45.5930289815... 7π
8 48.7107766217... 8π
9 51.7338428133... 9π
10 54.6752374468... 10π
11 57.5451651795... 11π
12 60.3518119691... 12π
13 63.1018679824... 13π
14 65.8008876380... 14π
15 68.4535449175... 15π

The choice of the index n is a bit crude. It is historically chosen in such a way that the index is 0 at the first value which is larger than the smallest positive zero (at imaginary part 14.13472515 ...) of the Riemann zeta function on the critical line. Notice, this -function oscillates for absolute-small real arguments and therefore is not uniquely invertible in the interval [-24,24]! Thus the odd theta-function has its symmetric Gram point with value 0 at index -3. Gram points are useful when computing the zeros of . At a Gram point ,

and if this is positive at two successive Gram points, must have a zero in the interval.

According to Gram’s law, the real part is usually positive while the imaginary part alternates with the gram points, between positive and negative values at somewhat regular intervals.

The number of roots, , in the strip from 0 to T, can be found by

where is an error term which grows asymptotically like .

Only if would obey Gram’s law, then finding the number of roots in the strip simply becomes

Today we know, that in the long run, Gram's law fails for about 1/4 of all Gram-intervals to contain exactly 1 zero of the Riemann zeta-function. Gram was afraid that it may fail for larger indices (the first miss is at index 126 before the 127th zero) and thus claimed this only for not too high indices. Later Hutchinson coined the phrase Gram's law for the (false) statement that all zeroes on the critical line would be separated by Gram points.

See also

References

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    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

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  • Wolfram Research – Riemann-Siegel Theta function (includes function plotting and evaluation)
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