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''<small>For a general overview of the theory, see [[General Relativity]]</small>''

''<small>For a general overview of the theory, see [[General Relativity]]</small>''

{{Template:Math-a}}

{{Math-a}}

General Relativity is a mathematical extension of Special Relativity. GR views space-time as a 4-dimensional [[manifold]], which looks locally like [[Minkowski space]], and which acquires [[curvature]] due to the presence of massive bodies. Thus, near massive bodies, the geometry of space-time differs to a large degree from [[Euclidean geometry]]: for example, the sum of the angles in a triangle is not exactly 180 degrees. Just as in classical physics, objects travel along [[geodesic]]s in the absence of external forces. Importantly though, near a massive body, geodesics are no longer straight lines. It is this phenomenon of objects traveling along geodesics in a curved spacetime that accounts for gravity.

General Relativity is a mathematical extension of Special Relativity. GR views space-time as a 4-dimensional [[manifold]], which looks locally like [[Minkowski space]], and which acquires [[curvature]] due to the presence of massive bodies. Thus, near massive bodies, the geometry of space-time differs to a large degree from [[Euclidean geometry]]: for example, the sum of the angles in a triangle is not exactly 180 degrees. Just as in classical physics, objects travel along [[geodesic]]s in the absence of external forces. Importantly though, near a massive body, geodesics are no longer straight lines. It is this phenomenon of objects traveling along geodesics in a curved spacetime that accounts for gravity.

Even at this level of examination, the fundamental thesis of the general theory of relativity is obvious: motion is determined by the curvature of spacetime, and the curvature of spacetime is determined by the matter, energy, momentum and fields within it.

Even at this level of examination, the fundamental thesis of the general theory of relativity is obvious: motion is determined by the curvature of spacetime, and the curvature of spacetime is determined by the matter, energy, momentum and fields within it.

===The right side of the equation: the stress-energy tensor===

==The right side of the equation: the stress-energy tensor==

In the [[Law of Universal Gravitation|Newtonian approximation]], the gravitational vector field is directly proportional to mass. In general relativity, mass is just one of several sources of spacetime curvature. The ''stress-energy tensor,'' <math>T_{\mu\nu}</math>, includes all of these sources. Put simply, the stress-energy tensor quantifies all the ''stuff'' that contributes to spacetime curvature, and thus to the gravitational field.

In the [[Law of Universal Gravitation|Newtonian approximation]], the gravitational vector field is directly proportional to mass. In general relativity, mass is just one of several sources of spacetime curvature. The ''stress-energy tensor,'' <math>T_{\mu\nu}</math>, includes all of these sources. Put simply, the stress-energy tensor quantifies all the ''stuff'' that contributes to spacetime curvature, and thus to the gravitational field.

In this way, we can see that the stress-energy tensor neatly quantifies ''all'' static and dynamic properties of a region of spacetime, from mass to momentum to electric charge to temperature to pressure to shear stress. Thus, the stress-energy tensor is all we need on the right-hand side of the equation in order to relate matter, energy and, well, ''stuff'' to curvature, and thus to the gravitational field.

In this way, we can see that the stress-energy tensor neatly quantifies ''all'' static and dynamic properties of a region of spacetime, from mass to momentum to electric charge to temperature to pressure to shear stress. Thus, the stress-energy tensor is all we need on the right-hand side of the equation in order to relate matter, energy and, well, ''stuff'' to curvature, and thus to the gravitational field.

====Example 1: Stress-energy tensor for a vacuum====

===Example 1: Stress-energy tensor for a vacuum===

The simplest possible stress-energy tensor is, of course, one in which all the values are zero.

The simplest possible stress-energy tensor is, of course, one in which all the values are zero.

In fact, it's not that simple. We'll discuss this in greater detail in the next section, but even a cursory qualitative examination can tell us there's more going on than that. Consider the gravitational field of an isolated body. A test particle placed somewhere near but outside of the body will move in a geodesic in spacetime, freely falling inward toward the central mass. A test particle with some constant linear velocity component perpendicular to the interval between the particle and the mass will move in a conic section. ''This is true even though the stress-energy tensor in that region is exactly zero.'' This much is obvious from our intuitive understanding of gravity: gravity affects things at a distance. But exactly ''how'' and ''why'' this happens, in the model of the Einstein field equations, is an interesting question which will be explored in [[#The left side of the equation: the Einstein curvature tensor|the next section]].

In fact, it's not that simple. We'll discuss this in greater detail in the next section, but even a cursory qualitative examination can tell us there's more going on than that. Consider the gravitational field of an isolated body. A test particle placed somewhere near but outside of the body will move in a geodesic in spacetime, freely falling inward toward the central mass. A test particle with some constant linear velocity component perpendicular to the interval between the particle and the mass will move in a conic section. ''This is true even though the stress-energy tensor in that region is exactly zero.'' This much is obvious from our intuitive understanding of gravity: gravity affects things at a distance. But exactly ''how'' and ''why'' this happens, in the model of the Einstein field equations, is an interesting question which will be explored in [[#The left side of the equation: the Einstein curvature tensor|the next section]].

====Example 2: Stress-energy tensor for an ideal dust====

===Example 2: Stress-energy tensor for an ideal dust===

Imagine a time-dependent distribution of identical, massive, non-interacting, electrically neutral particles. In general relativity, such a distribution is called a ''dust.'' Let's break down what this means.

Imagine a time-dependent distribution of identical, massive, non-interacting, electrically neutral particles. In general relativity, such a distribution is called a ''dust.'' Let's break down what this means.

We're now going to zoom out slightly from our model, such that we lose sight of the individual particles that make up our dust and can consider instead the dust as a whole. We can fully describe our dust at any ''event'' <math>P</math> — where ''event'' is defined as a point in space at an instant in time — by measuring the density <math>\rho</math> and the 4-velocity <math>u</math> at <math>P</math>. If we have those two pieces of information about the dust at every point within it at every moment in time, then there's literally nothing else to say about the dust: it's been fully described.

We're now going to zoom out slightly from our model, such that we lose sight of the individual particles that make up our dust and can consider instead the dust as a whole. We can fully describe our dust at any ''event'' <math>P</math> — where ''event'' is defined as a point in space at an instant in time — by measuring the density <math>\rho</math> and the 4-velocity <math>u</math> at <math>P</math>. If we have those two pieces of information about the dust at every point within it at every moment in time, then there's literally nothing else to say about the dust: it's been fully described.

=====Density=====

====Density====

Let's start by figuring out the density of dust at a the event <math>P</math>, as measured from the perspective of an observer moving along with the flow of dust at <math>P</math>. The density <math>\rho</math> is calculated very simply:

Let's start by figuring out the density of dust at a the event <math>P</math>, as measured from the perspective of an observer moving along with the flow of dust at <math>P</math>. The density <math>\rho</math> is calculated very simply:

Clearly proper density is a function of position, since it varies from point to point within the dust; the dust might be more "crowded" over here, less "crowded" over there. But it's also a function of time, because the configuration of the dust itself is time-dependent. If you measure the proper density at some point in space at one instant of time, then measure it at the same point in space at a different instant of time, you may get a different measurement. By convention, when dealing with a quantity that depends both on position in space and on time, physicists simply say that the quantity is a function of position, with the understanding that they're referring to a "position" in four-dimensional spacetime.

Clearly proper density is a function of position, since it varies from point to point within the dust; the dust might be more "crowded" over here, less "crowded" over there. But it's also a function of time, because the configuration of the dust itself is time-dependent. If you measure the proper density at some point in space at one instant of time, then measure it at the same point in space at a different instant of time, you may get a different measurement. By convention, when dealing with a quantity that depends both on position in space and on time, physicists simply say that the quantity is a function of position, with the understanding that they're referring to a "position" in four-dimensional spacetime.

=====4-velocity=====

====4-velocity====

The other quantity we need is ''4-velocity.'' Four-velocity is an extension of three-dimensional velocity (or 3-velocity). In three dimensional space, 3-velocity is a vector with three components. Likewise, in four-dimensional spacetime, 4-velocity is a vector with four components.

The other quantity we need is ''4-velocity.'' Four-velocity is an extension of three-dimensional velocity (or 3-velocity). In three dimensional space, 3-velocity is a vector with three components. Likewise, in four-dimensional spacetime, 4-velocity is a vector with four components.

Just as proper density is a function of position in spacetime, 4-velocity also depends on position. The 4-velocity of our dust at a given point in space won't necessarily be the same as the 4-velocity of the dust at another point in space. Likewise, the 4-velocity at a given point at a given time may not be the same as the 4-velocity of the dust at the same point at a different time. It helps to think of 4-velocity as the velocity of the dust ''through'' a point in both space and time.

Just as proper density is a function of position in spacetime, 4-velocity also depends on position. The 4-velocity of our dust at a given point in space won't necessarily be the same as the 4-velocity of the dust at another point in space. Likewise, the 4-velocity at a given point at a given time may not be the same as the 4-velocity of the dust at the same point at a different time. It helps to think of 4-velocity as the velocity of the dust ''through'' a point in both space and time.

=====Assembling the stress-energy tensor=====

====Assembling the stress-energy tensor====

Since the density and the 4-velocity fully describe our dust, we have everything we need to calculate the stress-energy tensor.

Since the density and the 4-velocity fully describe our dust, we have everything we need to calculate the stress-energy tensor.

From this equation, we can now calculate the contravariant components of the stress-energy tensor for an ideal dust.

From this equation, we can now calculate the contravariant components of the stress-energy tensor for an ideal dust.

======Time-time component======

=====Time-time component=====

We start with the contravariant time-time component <math>T^{00}</math>:

We start with the contravariant time-time component <math>T^{00}</math>:

Recall that <math>\rho</math> is a ''density'' quantity, in mass per unit volume. By the [[mass-energy equivalence]] principle, we know that <math>E = m c^2</math>. So we can interpret this component of the stress-energy tensor, which is written here in terms of mass-energy, to be equivalent to an ''energy density.''<ref>Actually rewriting the equation for the time-time component in terms of energy density requires refining our proper density equation into a form that doesn't depend on counting particles in a unit volume. Such a refinement is beyond the scope of this discussion. In less abstract dust solutions, the mass density is usually either assumed to be constant over space (as in the [[Friedmann-Lemaître-Robertson-Walker solution|FLRW solution]] that models a homogenous, isotropic expanding or contracting universe) or is assumed to depend only on the radius of the distribution (as in the [[LTB solution]] that models gravitational collapse). At this point, it is sufficient merely to understand that matter density and energy density, and matter flux and energy flux, are equivalent concepts under general relativity.</ref>

Recall that <math>\rho</math> is a ''density'' quantity, in mass per unit volume. By the [[mass-energy equivalence]] principle, we know that <math>E = m c^2</math>. So we can interpret this component of the stress-energy tensor, which is written here in terms of mass-energy, to be equivalent to an ''energy density.''<ref>Actually rewriting the equation for the time-time component in terms of energy density requires refining our proper density equation into a form that doesn't depend on counting particles in a unit volume. Such a refinement is beyond the scope of this discussion. In less abstract dust solutions, the mass density is usually either assumed to be constant over space (as in the [[Friedmann-Lemaître-Robertson-Walker solution|FLRW solution]] that models a homogenous, isotropic expanding or contracting universe) or is assumed to depend only on the radius of the distribution (as in the [[LTB solution]] that models gravitational collapse). At this point, it is sufficient merely to understand that matter density and energy density, and matter flux and energy flux, are equivalent concepts under general relativity.</ref>

======Off-diagonal components======

=====Off-diagonal components=====

The off-diagonal components of the tensor — <math>T^{\mu\nu}</math> where <math>\mu</math> and <math>\nu</math> are not equal — are calculated this way:

The off-diagonal components of the tensor — <math>T^{\mu\nu}</math> where <math>\mu</math> and <math>\nu</math> are not equal — are calculated this way:

In other words, in the case of an ideal dust, the stress-energy tensor is said to be ''symmetric.'' A rank two symmetric tensor is said to be symmetric if <math>T^{ab}=T^{ba}</math>.

In other words, in the case of an ideal dust, the stress-energy tensor is said to be ''symmetric.'' A rank two symmetric tensor is said to be symmetric if <math>T^{ab}=T^{ba}</math>.

======Diagonal space components======

=====Diagonal space components=====

The diagonal space components of the stress-energy tensor are calculated this way:

The diagonal space components of the stress-energy tensor are calculated this way:

So the diagonal space components of the stress-energy tensor come are expressed in terms of force per unit volume. Force per unit ''area'' are, of course, the traditional units of ''pressure'' in three-dimensional mechanics. So we can interpret the diagonal space components of the stress-energy tensor as the components of "4-pressure"<ref>Not a standard term.</ref> in spacetime.

So the diagonal space components of the stress-energy tensor come are expressed in terms of force per unit volume. Force per unit ''area'' are, of course, the traditional units of ''pressure'' in three-dimensional mechanics. So we can interpret the diagonal space components of the stress-energy tensor as the components of "4-pressure"<ref>Not a standard term.</ref> in spacetime.

======The big picture======

=====The big picture=====

We now know everything we know to assemble the entire stress-energy tensor, all sixteen components, and look at it as a whole.<ref>The stress-energy tensor is practically never written out in matrix form this way, even in textbooks. This is purely for illustration.</ref>

We now know everything we know to assemble the entire stress-energy tensor, all sixteen components, and look at it as a whole.<ref>The stress-energy tensor is practically never written out in matrix form this way, even in textbooks. This is purely for illustration.</ref>

As can easily be imagined, the task of constructing a stress-energy tensor for a system of ''arbitrary'' complexity can be a very daunting one. Fortunately, gravity is an extremely weak interaction, as interactions go, so on the scales where gravity is interesting, much of the complexity of a system can be approximated. For instance, there is absolutely nothing in the entire universe that behaves ''exactly'' like the ideal dust described here; every massive particle interacts, in one way or another, with other massive particles. No matter what, a real system is going to be ''very'' much more complex than this approximation. Yet, the ideal dust solution remains a much-used approximation in theoretical physics specifically ''because'' gravity is such a weak interaction. On the scales where gravity is worth studying, many distributions of matter, including interstellar nebulae, clusters of galaxies, even the whole universe really do behave very much like an ideal dust.

As can easily be imagined, the task of constructing a stress-energy tensor for a system of ''arbitrary'' complexity can be a very daunting one. Fortunately, gravity is an extremely weak interaction, as interactions go, so on the scales where gravity is interesting, much of the complexity of a system can be approximated. For instance, there is absolutely nothing in the entire universe that behaves ''exactly'' like the ideal dust described here; every massive particle interacts, in one way or another, with other massive particles. No matter what, a real system is going to be ''very'' much more complex than this approximation. Yet, the ideal dust solution remains a much-used approximation in theoretical physics specifically ''because'' gravity is such a weak interaction. On the scales where gravity is worth studying, many distributions of matter, including interstellar nebulae, clusters of galaxies, even the whole universe really do behave very much like an ideal dust.

===The left side of the equation: the Einstein curvature tensor===

==The left side of the equation: the Einstein curvature tensor==

We will recall that the Einstein field equations can be written as a single tensor equation:

We will recall that the Einstein field equations can be written as a single tensor equation:

The left side of the equation, then, is the "space" side. Matter tells space how to curve, and space tells matter how to move. So the left side of the Einstein field equation must necessarily describe the curvature of spacetime in the presence of matter and energy.

The left side of the equation, then, is the "space" side. Matter tells space how to curve, and space tells matter how to move. So the left side of the Einstein field equation must necessarily describe the curvature of spacetime in the presence of matter and energy.

====Some assumptions about the universe====

===Some assumptions about the universe===

Before we proceed into a discussion of what curvature is and how the Einstein equation describes it, we must first pause to state some fundamental assumptions about the universe.<ref>As we will later see, these assumptions may in fact turn out not to be valid for all of spacetime. It may be more accurate, although significantly less satisfying, to say that we assume these things to be true about spacetime, ''except where they aren't.''</ref>

Before we proceed into a discussion of what curvature is and how the Einstein equation describes it, we must first pause to state some fundamental assumptions about the universe.<ref>As we will later see, these assumptions may in fact turn out not to be valid for all of spacetime. It may be more accurate, although significantly less satisfying, to say that we assume these things to be true about spacetime, ''except where they aren't.''</ref>

With these two assumptions and this convenient property in hand, we will now examine what it means to say that spacetime is curved.

With these two assumptions and this convenient property in hand, we will now examine what it means to say that spacetime is curved.

====Flatness versus curvature====

===Flatness versus curvature===

Let's start by considering the simplest possible geometry<ref>Well, the simplest possible ''interesting'' geometry, anyway.</ref>: the [[Euclidean plane]].

Let's start by considering the simplest possible geometry:<ref>Well, the simplest possible ''interesting'' geometry, anyway.</ref> the [[Euclidean plane]].

The Euclidean plane is an infinite, flat, two-dimensional surface. A sheet of paper is a good approximation of the Euclidean plane. Onto this plane, we can project a set of [[Cartesian coordinate system|Cartesian coordinates]]. By "Cartesian," we mean that the coordinate axes are straight lines, that they are perpendicular, and that the unit lengths of the axes are equal. A fancier term for a Cartesian coordinate system is an ''orthonormal basis.''

The Euclidean plane is an infinite, flat, two-dimensional surface. A sheet of paper is a good approximation of the Euclidean plane. Onto this plane, we can project a set of [[Cartesian coordinate system|Cartesian coordinates]]. By "Cartesian," we mean that the coordinate axes are straight lines, that they are perpendicular, and that the unit lengths of the axes are equal. A fancier term for a Cartesian coordinate system is an ''orthonormal basis.''

But this brings up an important point. The ''entire'' surface of the sphere is curved, and thus can't be approximated with Cartesian coordinates. But a sufficiently ''small'' patch of the surface can be approximated with Cartesian coordinates. This implies, then, that "curvedness" isn't an either-or property. Somewhere between the locally flat region of the surface and the entire surface, the ''amount'' of curvature goes from none to some value. Curvature, then, must be something we can measure.

But this brings up an important point. The ''entire'' surface of the sphere is curved, and thus can't be approximated with Cartesian coordinates. But a sufficiently ''small'' patch of the surface can be approximated with Cartesian coordinates. This implies, then, that "curvedness" isn't an either-or property. Somewhere between the locally flat region of the surface and the entire surface, the ''amount'' of curvature goes from none to some value. Curvature, then, must be something we can measure.

====The metric tensor====

===The metric tensor===

It is a fundamental property of the Euclidean plane that, when Cartesian coordinates are used, the distance <math>s</math> between any two points <math>A</math> and <math>B</math> is given by the following equation:

It is a fundamental property of the Euclidean plane that, when Cartesian coordinates are used, the distance <math>s</math> between any two points <math>A</math> and <math>B</math> is given by the following equation:

But relativity is concerned not with geometrically abstract ''space;'' we're interested in very real space''time,'' and that requires a slightly different kind of metric.

But relativity is concerned not with geometrically abstract ''space;'' we're interested in very real space''time,'' and that requires a slightly different kind of metric.

====The local Minkowski metric====

===The local Minkowski metric===

====Geodesics====

===Geodesics===

====Parallel transport and intrinsic curvature====

===Parallel transport and intrinsic curvature===

====The Riemann and Ricci tensors and the curvature scalar====

===The Riemann and Ricci tensors and the curvature scalar===

====The Einstein tensor====

===The Einstein tensor===

====The cosmological constant====

===The cosmological constant===

{{Template:Relativity}}

{{Relativity}}

 

{{Reflist}}