# User:Ems57fcva/sandbox/General Relativity

## Introduction

General relativity (GR) or general relativity theory (GRT) is the theory of gravitation published by Albert Einstein in 1915. GR is a geometrical theory under which spacetime is a 4-dimensional 3+1 pseudo-Riemannian manifold that is curved by the mass, energy, and momentum (or "substance") within it.

In general relativity, the relationship between substance and curvature is governed by the Einstein Field Equations. The solutions to these field equations are metrics of spacetime which describe the spacetime and from which the geodesic equations of motion are obtained.

GR was developed by Einstein staring in 1907 with the publication of the Principle of Equivalence, which describes gravitation and acceleration as different perspectives of the same thing. An important consequence of General Relativity and its Equivalence Principle is that free-fall is inertial motion, while being "at rest" on the surface of the Earth is actually an accelerated (or non-inertial) state.

Although it was the start of Einstein's investigations, the Equivalence Principle is not the underlying principle of GR. Instead, it is now a consequence of GR, the theory build to explain it. The full set of underlying principle for GR are

• The General Principle of Relativity: The laws of physics are the same in all frames of reference
• The Principle of General Covariance: The laws of physics are independent of the coordinate system in which they are expressed.
• Local Lorentz Invariance: The rules of special relativity (or SR) apply locally in all frames of reference.
• The Principle of geodesic motion: Inertial motion occurs along timelike geodesics as parameterized by proper time.
• Spacetime is a pseuo-Rienmanian manifold curved by the substance within it.

## What is inertia?

To understand what GR is about, one needs to realize that one of Einstein's goals was to answer this seemingly simple question. In this section, we will review the concept of inertia and show it is affected by the various principles of GR mentioned above.

### First try: Newton's First Law

Newton's first law of motion is

An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

This may sound good, but what is the meaning of "in the same direction"? For example, let us look at an object going in straight line first against a Cartesian coordiante system (as showm in fugure 1A), and then against a radial coordinate system (as shown in figure 1B). In the Cartesian case, the object really is going in the same direction as defined by the coordinate system, but in the radial coordinate case, the object is first coming mostly inward, then is going tangetially, and then is heading outward! Yet both trajectories for the object are the same: All that has changes is how the space through which it is traveling is mapped.

What Einstein realized was that space and time simply are, and that the coordinate system is a artifical mathematical aid added in to help describe actions and interactions with the spacetime. So in trying to describe Newton's First Law, we come up againt the first of the underlying concepts of General Relativity, the Principle of General Covariance:

The laws of physics are the same in all coordinate systems.

We now need a concept that unifies how motion is described in both figures 1A and 1B. To do this, we need turn to the work of Riemann and others on non-Euclidean geometries. Through their work, the concept of a geodesic path came into being. In a Euclidean (or flat) manifold, these geodesics turn out to be straight lines. With the concept of a geodesic in place, we now need to define the manifolds for space and time in Newtonian Mechanics. These are:

• Space is a three-dimensional Euclidean Riemannian manifold, and
• Time is a one-dimensional manifold which proceeds at the same rate at all positions and for all observers.

Now the first law of motion can be written as

A object will move along a geodesic path in space as parameterized by distance travelled at a constant rate with repect to time unless it should be acted on by an unbalanced force.

So inertial motion is now defined as movement along a geodesic path. This is a first draft of another fundamental principle of General Relativity: the Principle of Geodesic Motion. We will revisit and revise this principle soon.

### Second try: Special Relativity

After the introduction of SR by Eintein in 1905, the geometric structure of its spacetime was described by Minkowski in 1908. In a landmark talk, he noted that in SR, there is an "invariant distance" between events ${\displaystyle s}$that all observers will agree on. In a cartesian coordinate system with a time coordinate ${\displaystyle t}$ and linear orthogonal spatial coordinates ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ the formula for this invariant distance is

As it tunrs out, ${\displaystyle s}$ is the elapsed proper time for an observer moving linearly between the events (when this is possible). In addition, the cases of linear motion between events in SR represents the geodesics of this spacetime in a timelike direction.

Another feature of this Minkowski spacetime of GR is that coordinate rest, which was permitted in Newton's physics, is missing in relativity: Now with the unification of space and time one is always in some sort of coordinate motion since one is always moving forward through time.

Through these effects, we find that a refinement is needed to our above stated rules for inertia. Now

An object will move though spacetime along a timelike geodesic of spacetime as parameterized by proper time unless it is acted on by an unbalanced force.

This is the definition of inertial motion that is used in relativity.

### Third try: The Equivalence Principle

It is all fine and dandy to have a definition of inertial motion, but how does one know that they are in an inertial frame of reference?

## Verification of General Relativity

=== The orbit of Mercury ===woob doob

when light ray

### Binary Stars and Pulsars

+++ END OF EDITTING +++

The meaning of the Principle of Equivalence has gradually broadened, in consonance with Einstein's further writings, to include the concept that no physical measurement within a given unaccelerated reference system can determine its state of motion. This implies that it is impossible to measure, and therefore virtually meaningless to discuss, changes in fundamental physical constants, such as the rest masses or electrical charges of elementary particles in different states of relative motion. Any measured change in such a constant would represent either experimental error or a demonstration that the theory of relativity was wrong or incomplete.

This principle explains the experimental observation that inertial and gravitational mass are equivalent. Moreover, the principle implies that some frames of reference must obey a non-Euclidean geometry: that spacetime is curved (by matter and energy), and gravity can be seen purely as a result of this geometry. This yields many predictions such as gravitational redshifts and light bending around stars, black holes, time slowed by gravitational fields, and slightly modified laws of gravitation even in weak gravitational fields. However, it should be noted that the equivalence principle does not uniquely determine the field equations of curved spacetime, and there is a parameter known as the cosmological constant which can be adjusted.

The modifications to Isaac Newton's law of universal gravitation produced the first great theoretical success of general relativity: the correct prediction of the precession of the perihelion of Mercury's orbit. Many other quantitative predictions of general relativity have since been confirmed by astronomical observations. However, because of the difficulty in making these observations, theories which are similar but not identical to general relativity, such as the Brans-Dicke theory and the Rosen bi-metric theory cannot be ruled out completely, and current experimental tests can be viewed as reducing the deviation from GR which is allowable. However, the discovery in 2003 of PSR J0737-3039, a binary neutron star in which one component is a pulsar and where the perihelion precesses 16.88° per year (or about 140,000 times faster than the precession of Mercury's perihelion), enabled the most precise experimental verification yet of effects predicted by general relativity. [1] [2]

There are no known experimental results that suggest that a theory of gravity radically different from general relativity is necessary. For example, the Allais effect was initially speculated to demonstrate "gravitational shielding," but was subsequently explained by conventional phenomena.

Nevertheless, there are good theoretical reasons for considering general relativity to be incomplete. General relativity does not include quantum mechanics, and this causes the theory to break down at sufficiently high energies. A continuing unsolved challenge of modern physics is the question of how to correctly combine general relativity with quantum mechanics, thus applying it also to the smallest scales of time and space.

## The curvature of spacetime

Mathematicians use the term "curved" to refer to any space whose geometry is non-Euclidean. Frequently, this curvature is illustrated by an image like the one below.

Two-dimensional visualisation of space-time distortion

This image represents spacetime as a higher-dimensional flat space, with the "weight" of a massive object "stretching" the trampoline-like spacetime "fabric", which would result in trajectories around this "dent" being curved due to the "slope" and the pull of gravity in some higher dimension . This image, however, is only suggestive of the reality. It is important to remember that spacetime is curved, not merely space, and that space is three-dimensional, not two-dimensional as shown.

Another approach used to understand spacetime as a curved surface in three-dimensional space is to instead begin by imagining a universe of one-dimensional beings living in one dimension of space and one dimension of time. Each bit of matter is not a point on whatever curved surface you imagine, but a line showing where that point moves as it goes from the past to the future. These lines are called world lines.

While it can be helpful for visualization to imagine a curved surface sitting in space of a higher dimension, that model is not very useful for the real universe; although a two dimensional surface can be embedded in three, and thus visualized well, a curved four dimensional spacetime such as our universe cannot be imbedded in a flat space of even five dimensions, but many more are required. Curvature can be measured entirely within a surface, and similarly within a higher-dimensional manifold such as space or spacetime. On earth, if you start at the North Pole, walk south for about 10,000 km (to the Equator), turn left by 90 degrees, walk for 10,000 more km, and then do the same again (walk for 10,000 more km, turn left by 90 degrees, walk for 10,000 more km), you will be back where you started. Such a triangle with three right angles is only possible because the surface of the earth is curved. The curvature of spacetime can be evaluated, and indeed given meaning, in a similar way. Spaces of only two dimensions, however, require only one quantity, the Gaussian or scalar curvature, to quantify their curvature. In more dimensions, curvature is quantified by the Riemann tensor. This tensor describes how a vector that is moved along a curve parallel to itself changes when a round trip is made. In flat space the vector returns to the same orientation, but in a curved space it generally does not.

## Relationship to special relativity

The special theory of relativity (1905) modified the equations used in comparing the measurements made by differently moving bodies, in view of the constant value of the speed of light, i.e. its observed invariance in reference frames moving uniformly relative to each other. This had the consequence that physics could no longer treat space and time separately, but only as a single four-dimensional system, "space-time," which was divided into "time-like" and "space-like" directions differently depending on the observer's motion. The general theory added to this that the presence of matter "warped" the local space-time environment, so that apparently "straight" lines through space and time have the properties we think of "curved" lines as having.

On May 29, 1919, observations by Arthur Eddington of shifted star positions during a solar eclipse confirmed the theory.

## Foundations

General relativity's mathematical foundations go back to the axioms of Euclidean geometry and the many attempts over the centuries to prove Euclid's fifth postulate, that parallel lines remain always equidistant, culminating with the realisation by Lobachevsky, Bolyai and Gauss that this axiom need not be true. The general mathematics of non-Euclidean geometries was developed by Gauss' student, Riemann, but these were thought to be mostly inapplicable to the real world until Einstein developed his theory of relativity. The existing applications were restricted to the geometry of curved surfaces in Euclidean space, as if one lived and moved in such a surface. While such applications seem trivial compared to the calculations in the four dimensional spacetimes of general relativity, they provided a minimal development and test environment for some of the equations.

Gauss had realised that there is no a priori reason that the geometry of space should be Euclidean. What this means is that if a physicist holds up a stick, and a cartographer stands some distance away and measures its length by a triangulation technique based on Euclidean geometry, then he is not guaranteed to get the same answer as if the physicist brings the stick to him and he measures its length directly. Of course for a stick he could not in practice measure the difference between the two measurements, but there are equivalent measurements which do detect the non-Euclidean geometry of space-time directly; for example the Pound-Rebka experiment (1959) detected the change in wavelength of light from a cobalt source rising 22.5 meters against gravity in a shaft in the Jefferson Physical Laboratory at Harvard, and the rate of atomic clocks in GPS satellites orbiting the Earth has to be corrected for the effect of gravity.

Newton's theory of gravity had assumed that objects did in fact have absolute velocities: that some things really were at rest while others really were in motion. He realized, and made clear, that there was no way these absolutes could be measured. All the measurements one can make provide only velocities relative to one's own velocity (positions relative to one's own position, and so forth), and all the laws of mechanics would appear to operate identically no matter how one was moving. Newton believed, however, that the theory could not be made sense of without presupposing that there are absolute values, even if they cannot be determined. In fact, Newtonian mechanics can be made to work without this assumption: the outcome is rather innocuous, and should not be confused with Einstein's relativity which further requires the constancy of the speed of light.

In the nineteenth century, Maxwell formulated a set of equations—Maxwell's field equations—that demonstrated that electromagnetic fields behave as waves travelling at, amazingly, the speed of light in a vacuum; hence, the identification of light with electromagnetic fields was made. This appeared to provide a way around Newton's relativity: by comparing one's own speed with the speed of light in one's vicinity, one should be able to measure one's absolute speed--or, what is practically the same, one's speed relative to a frame of reference that would be the same for all observers.

The assumption was made that whatever medium light was travelling through—whatever it was waves of—could be treated as a background against which to make other measurements. This inspired a search to determine the earth's velocity through this cosmic backdrop or "aether"—the "aether drift." The speed of light measured from the surface of the earth should appear to be greater when the earth was moving against the aether, slower when they were moving in the same direction. (Since the earth was hurtling through space and spinning, there should be at least some regularly changing measurements here.) A test made by Michelson and Morley toward the end of the century had the astonishing result that the speed of light appeared to be the same in every direction.

In his 1905 paper "On the Electrodynamics of Moving Bodies", Einstein explained these results in his theory of special relativity.

## Outline of the theory

The fundamental idea in relativity is that we cannot talk of the physical quantities of velocity or acceleration without first defining a reference frame, and that a reference frame is defined by choosing particular matter as the basis for its definition. Thus all motion is defined and quantified relative to other matter. In the special theory of relativity it is assumed that reference frames can be extended indefinitely in all directions in space and time. The theory of special relativity concerns itself with inertial (non-accelerating) frames while general relativity deals with all frames of reference. In the general theory it is recognised that we can only define local frames to given accuracy for finite time periods and finite regions of space (similarly we can draw flat maps of regions of the surface of the earth but we cannot extend them to cover the whole surface without distortion). In general relativity Newton's laws are assumed to hold in local reference frames. In particular free particles travel in straight lines in local inertial (Lorentz) frames. When these lines are extended they do not appear straight, and are known as geodesics. Thus Newton's first law is replaced by the law of geodesic motion.

We distinguish inertial reference frames, in which bodies maintain a uniform state of motion unless acted upon by another body, from non-inertial frames in which freely moving bodies have an acceleration deriving from the reference frame itself. In non-inertial frames there is a perceived force which is accounted for by the acceleration of the frame, not by the direct influence of other matter. Thus we feel acceleration when cornering on the roads when we use a car as the physical base of our reference frame. Similarly there are coriolis and centripetal forces when we define reference frames based on rotating matter (such as the Earth or a child's roundabout). The principle of equivalence in general relativity states that there is no local experiment to distinguish non-rotating free fall in a gravitational field from uniform motion in the absence of a gravitational field. In short there is no gravity in a reference frame in free fall. From this perspective the observed gravity at the surface of the Earth is the force observed in a reference frame defined from matter at the surface which is not free, but is acted on from below by the matter within the Earth, and is analogous to the acceleration felt in a car.

Mathematically, Einstein models space-time by a four-dimensional pseudo-Riemannian manifold, and his field equation states that the manifold's curvature at a point is directly related to the stress energy tensor at that point; the latter tensor being a measure of the density of matter and energy. Curvature tells matter how to move, and matter tells space how to curve.

The field equation is not uniquely proven (it is only an assumption of GR) and there is room for other models, provided that they do not contradict observation. General relativity is distinguished from other theories of gravity by the simplicity of the coupling between matter and curvature, although we still await the unification of general relativity and quantum mechanics and the replacement of the field equation with a deeper quantum law. Few physicists doubt that such a theory of everything will give general relativity in the appropriate limit, just as general relativity predicts Newton's law of gravity in the non-relativistic limit.

Einstein's field equation contains a parameter called the "cosmological constant" ${\displaystyle \Lambda }$ which was originally introduced by Einstein to allow for a static universe (i.e., one that is not expanding or contracting). This effort was unsuccessful for two reasons: the static universe described by this theory was unstable, and observations by Hubble a decade later confirmed that our universe is in fact not static but expanding. So ${\displaystyle \Lambda }$ was abandoned, with Einstein calling it the "biggest blunder [I] ever made". However, quite recently, improved astronomical techniques have found that a non-zero value of ${\displaystyle \Lambda }$ is needed to explain some observations.

### Einstein field equation

The field equation reads, in components, as follows:

${\displaystyle R_{ab}-{R \over 2}g_{ab}+\Lambda g_{ab}={8\pi G \over c^{4}}T_{ab}}$

where ${\displaystyle R_{ab}}$ are the Ricci curvature tensor components, ${\displaystyle R}$ is the scalar curvature, ${\displaystyle g_{ab}}$ are the metric tensor components, ${\displaystyle \Lambda }$ is the cosmological constant, ${\displaystyle T_{ab}}$ are the stress-energy tensor components describing the non-gravitational matter, energy and forces at any given point in space-time, ${\displaystyle \pi }$ is pi, ${\displaystyle c}$ is the speed of light in a vacuum and ${\displaystyle G}$ is the gravitational constant which also occurs in Newton's law of gravity.

The Ricci tensor and scalar curvature are themselves derivable from the metric, which describes the metric of the manifold and is a symmetric 4 x 4 tensor, so it has 10 independent components. Given the freedom of choice of the four spacetime coordinates, the independent equations that make up the Einstein field equation reduce to 6.

Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side, written as part of the stress-energy tensor, and then interpreted as a form of dark energy whose density is constant in space-time.

The study of the solutions of this equation is one of the activities of cosmology. It leads to the prediction of black holes and to different models of evolution of the universe.

Solutions of the field equations are sometimes known as "metrics" or "spacetimes". Some well-known and popular metrics include:

1. Schwarzschild metric (which describes the spacetime geometry around a spherical mass)
2. Kerr metric (which describes the geometry around a rotating spherical mass)
3. Reissner-Nordstrom metric (which describes the geometry around a charged spherical mass)
4. Kerr-Newman metric (which describes the geometry around a charged-rotating spherical mass)
5. Friedmann-Robertson-Walker (FRW) metric (which is an important model of an expanding universe)
6. pp-wave metrics (which describe various types of gravitational waves)
7. wormhole metrics (which serve as theoretical models for time travel)
8. Alcubierre metric (which serves as a theoretical model of space travel)

Solutions (1), (2), (3) and (4) also include black holes.

## The vierbein formulation of general relativity

This is an alternative equivalent formulation of general relativity using four reference vector fields, called a vierbein or tetrad. We have four vector fields, ea, a = 0, 1, 2, 3 such that g(ea, eb) = ηab where

${\displaystyle \eta ={\begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}}}$.

See sign convention. One thing to note is that we can perform an independent proper, orthochronous Lorentz transformation at each point (subject to smoothness, of course) and still get a valid tetrad. So, the tetrad formulation of GR is a gauge theory, but with a noncompact gauge group SO(3,1). It is also invariant under diffeomorphisms.

See vierbein and Palatini action for more details. See Einstein-Cartan theory for an extension of general relativity to include torsion. See teleparallelism for another theory which predicts the same results as general relativity but with FLAT spacetime (no curvature).

## Quotes

The theory appeared to me then, and still does, the greatest feat of human thinking about nature, the most amazing combination of philosophical penetration, physical intuition, and mathematical skill. But its connections with experience were slender. It appealed to me like a great work of art, to be enjoyed and admired from a distance.Max Born

## References

### Textbooks

• Carroll, Sean M., Spacetime and Geometry: An introduction to general relativity, Addison Wesley, San Francisco (2004). ISBN 0-8053-8732-3. A modern graduate level textbook.
• D'Inverno, Ray, Introducing Einstein's Relativity, Oxford University Ass Press (1993). A modern undergraduate level text.
• Misner, Charles, Kip Thorne, and John Wheeler, Gravitation, Freeman (1973). ISBN 0716703440. A classic graduate level text book, which, if somewhat long winded, pays more attention to the geometrical basis and the development of ideas in general relativity than some other approaches.

### Other

• Bondi, Herman, Relativity and Common Sense, Heinemann (1964). A school level introduction to the principle of relativity by a renowned scientist.
• Einstein, Albert, Relativity: The special and general theory. ISBN 0517884410. The special and general relativity theories in their original form.
• Epstein, Lewis Caroll, Relativity Visualized. ISBN 093521805X. Requires no mathematical background. Actually explains general relativity, rather than merely hinting at it with a few metaphors.
• Perret, W. and G.B. Jeffrey, trans.: The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity, New York Dover (1923).
• Thorne, Kip, and Stephen Hawking, Black Holes and Time Warps, Papermac (1995). A recent popular account by leading experts.
• J. J. O'Connor and E. F. Robertson, History of General Relativity at the MacTutor History of Mathematics archive.
• The original 1915 article by David Hilbert containing the gravitational field equation.
• Malcolm MacCallum's GR News service for current research in relativity.