2009-11-15T02:47:00Z

KSorenson: No need to single anybody out, I don't think.

{{Progress}}

The '''General Theory of Relativity''' is an extension of [[Special theory of relativity|Special Relativity]], dealing with curved coordinate systems, accelerating frames of reference, curvilinear motion, and curvature of spacetime itself. It could be said that General Relativity is to Special Relativity as vector calculus is to vector algebra. General Relativity is best known for its formulation of gravity as a [[fictitious force]] arising from the curvature of spacetime. In fact, "General Relativity" and "Einstein's formulation of gravity" are nearly synonymous in many people's minds.

The General Theory of Relativity was first published by [[Albert Einstein]] in 1916.

General Relativity, like [[Quantum Mechanics]] (the other of the two theories comprising "modern physics") both have reputations for being notoriously complicated and difficult to understand. In fact, in the early decades of the 20th century, General Relativity had a sort of cult status in this regard. General Relativity and Quantum Mechanics are both advanced college-level and postgraduate level topics. No encyclopedia article can really explain either of them fully. But, in the article below, we will attempt to give a rough outline of the General Relativistic formulation of gravity.

The general theory of relativity is a ''metric theory,'' sometimes also called a ''geometric theory.'' Metric theories describe physical phenomena in terms of [[differential geometry]]. This stands in contrast to Isaac Newton's [[Law of Universal Gravitation|law of universal gravitation]], which described gravity in terms of a [[vector field]]. In the case of general relativity, the theory relates ''stress-energy'' — an extension of the concept of [[mass]] — and the [[curvature]] of [[spacetime]]. In the words of physicist John Wheeler, "Space tells matter how to move, matter tells space how to curve."<ref>Misner, Thorne & Wheeler. ''Gravitation.'' (1973)</ref>

In the [[weak field approximation]], where velocities of moving objects are low and gravitational fields are not very severe, the theory of general relativity is said to ''reduce to'' the law of universal gravitation. That is to say, under those circumstances the equations of general relativity are mathematically equivalent to the equations of Newtonian gravitation.

The theory was inspired by a thought experiment developed by Einstein involving two elevators. The first elevator is stationary on the Earth, while the other is being pulled through space at a constant acceleration of g. Einstein realized that any physical experiment carried out in the elevators would give the same result. This realization is known as the equivalence principle and it states that accelerating frames of reference and gravitational fields are indistinguishable. General Relativity is the theory of gravity that incorporates Special Relativity and the equivalence principle.

==Coordinate Systems and Spacetime Diagrams==

... In progress ...

==Qualitative Introduction to General Relativity==

The relationship between the curvature of spacetime and the motions of freely falling bodies is often explained by an easily imagined analogy: bowling balls and golf balls on a trampoline.

Imagine that we place a golf ball on an ordinary backyard trampoline. If we give the golf ball a slight push, it will roll along in a straight line until friction brings it to a halt. But if we imagine that friction doesn't exist, then the golf ball will roll in a straight line at a constant speed forever — or at least until it reaches the edge of the trampoline and falls off.

Now imagine a bowling ball sitting in the middle of a trampoline. The trampoline isn't a rigid surface, so it deforms where the weight of the bowling ball pushes it down. This causes the surface to be curved downward, toward the ground.

If we place a golf ball near the edge of the trampoline, it will begin to roll toward the bowling ball, because the trampoline is sloped downward in that direction. The golf ball will start off moving very slowly, then pick up speed as it approaches the bowling ball, until finally it collides with the bowling ball and comes to rest.

But if we give the golf ball a slight push in a direction perpendicular to the direction of the bowling ball, then it will move in a curved path. If we only push it a little bit, the golf ball will curve slightly, but still collide with the bowling ball. If we push the golf ball somewhat harder, it will curve toward the bowling ball, pass by it on one side and climb back out of the depression until it reaches the edge and falls off.

But if we're ''very'' careful, and give the golf ball ''just the right'' push, it will curve completely around the bowling ball and return to our hand.

This is, in a nutshell, how spacetime and matter interact under the theory of general relativity. Massive objects — represented in our analogy by the bowling ball — curve spacetime. Less-massive objects also curve spacetime, but to a lesser extent. If the object is small enough, like our golf ball, the amount of curvature is so slight that we can't even measure it.

The way the golf ball moved in the three scenarios we imagined correspond to ''conic-section orbits,'' or ''Kepler orbits.'' When we just placed the golf ball and it rolled straight toward the bowling ball, that was a ''degenerate'' orbit: a straight line. When we gave it a push and it curved around the bowling ball and off the edge of the trampoline, that was a ''hyperbolic'' orbit. And when we gave it just the right push so that it curved around the bowling ball and back to our hand, that was an ''elliptical'' orbit.

These are the same orbits that are predicted by Isaac Newton's [[Law of Universal Gravitation|law of universal gravitation]]. But in Newton's equations, objects move in conic-section orbits because of a force that accelerates them toward the central mass. In general relativity, objects move in conic-section orbits because spacetime itself is curved, just like our imaginary trampoline was curved by the bowling ball.

Of course, our analogy is far from perfect. Our imaginary trampoline curved ''downward,'' toward the ground, pushed down by the weight of the bowling ball. That's not how spacetime behaves in general relativity. It curves, but not ''toward'' anything, not in any ''direction.'' Spacetime in general relativity is instead said to have ''intrinsic curvature,'' which is mathematically quite simple but very difficult to visualize.

And of course there are many, many other aspects of general relativity that our imaginary trampoline didn't model. But the analogy captures the essential nature of the theory: the bowling ball caused the trampoline to be curved, and the curvature of the trampoline caused the golf ball to move in a different way than if the trampoline had been flat. This is the essence of general relativity: matter tells space how to curve, and space tells matter how to move.

==Quantitative Introduction to General Relativity==

{{Template: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.

The mathematical expression of the theory of general relativity takes the form of the ''Einstein field equations,'' a set of ten nonlinear partial differential equations. While solving these equations is quite difficult, examining them provides valuable insight into the structure and meaning of the theory.

In their general form, the Einstein field equations are written as a single [[tensor]] equation in [[abstract index notation]] relating the curvature of spacetime to sources of curvature such as energy density and momentum.

:<math> G_{\mu\nu} = 8\,\pi\,G\,T_{\mu\nu} </math>

In this form, <math>G_{\mu\nu}</math> represents the ''Einstein tensor,'' <math>G</math> is the same ''gravitational constant'' that appears in the [[Law of Universal Gravitation|law of universal gravitation]], and <math>T_{\mu\nu}</math> is the ''stress-energy tensor'' (sometimes referred to as the ''energy-momentum tensor).'' The indices <math>\mu</math> and <math>\nu</math> range from zero to three, representing the time coordinate and the three space coordinates in a manner consistent with [[Special theory of relativity|special relativity]].

The left side of the equation — the Einstein tensor — describes the curvature of spacetime in the region under examination. The right side of the equation describes everything in that region that affects the curvature of spacetime.

As we can clearly see even in this simplified form, the Einstein field equations can be solved "in either direction." Given a description of the gravitating matter, energy, momentum and fields in a region of spacetime, we can calculate the curvature of spacetime surrounding that region. On the other hand, given a description of the curvature of a region spacetime, we can calculate the motion of a test particle anywhere within that region.

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===

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.

First we will define the stress-energy tensor technically, then we'll examine what that definition means. In technical terms, the stress energy tensor represents ''the flux of the <math>\mu</math> component of 4-momentum across a surface of constant coordinate <math>x^\nu</math>.''

Fine. But what does that ''mean?''

In classical mechanics, it's customary to refer to coordinates in space as <math>x</math>, <math>y</math> and <math>z</math>. In general relativity, the convention is to talk instead about coordinates <math>x^0</math>, <math>x^1</math>, <math>x^2</math>, and <math>x^3</math>, where <math>x^0</math> is the time coordinate otherwise called <math>t</math>, and the other three are just the <math>x</math>, <math>y</math> and <math>z</math> coordinates. So "a surface of constant coordinate "<math>x^\nu</math>" simply means a 3-plane perpendicular to the <math>x^\nu</math> axis.

The ''flux'' of a quantity can be visualized as the magnitude of the current in a river: the flux of water is the amount of water that passes through a cross-section of the river in a given interval of time. So more generally, the flux of a quantity across a surface is the amount of that quantity that passes through that surface.

''Four-momentum'' is the special relativity analogue of the familiar momentum from classical mechanics, with the property that the time coordinate <math>\mathbf{P}^0</math> of a particle's four-momentum is simply the energy of the particle; the other three components of four-momentum are the same as in classical momentum.

So putting that all together, the stress-energy tensor is the flux of 4-momentum across a surface of constant coordinate. In other words, the stress-energy tensor describes the density of energy and momentum, and the flux of energy and momentum in a region. Since under the [[mass-energy equivalence]] principle we can convert mass units to energy units and vice-versa, this means that the stress-energy tensor describes ''all the mass and energy in a given region of spacetime.''

Put even more simply, the stress-energy tensor represents ''everything that gravitates.''

The stress-energy tensor, being a tensor of rank two in four-dimensional spacetime, has sixteen components that can be written as a 4 × 4 matrix.

:<math>T_{\mu \nu }=\begin{pmatrix}
\color{Purple}T_{00} & \color{Blue}T_{01} & \color{Blue}T_{02} & \color{Blue}T_{03} \\
\color{Red}T_{10} & \color{Orange}T_{11} & \color{Green}T_{12} & \color{Green}T_{13} \\
\color{Red}T_{20} & \color{Green}T_{21} & \color{Orange}T_{22} & \color{Green}T_{23} \\
\color{Red}T_{30} & \color{Green}T_{31} & \color{Green}T_{32} & \color{Orange}T_{33}
\end{pmatrix}</math>

Here the components have been color-coded to help clarify their physical interpretations.

:;<math>\color{Purple}T_{00}</math>
::energy density, which is equivalent to mass-energy density; this component includes the mass contribution

:;<math>\color{Red}T_{10}</math>, <math>\color{Red}T_{20}</math>, <math>\color{Red}T_{30}</math>
::the components of momentum density

:;<math>\color{Blue}T_{01}</math>, <math>\color{Blue}T_{02}</math>, <math>\color{Blue}T_{03}</math>
::the components of energy flux

The space-space components of the stress-energy tensor are simply the [[stress tensor]] from classic mechanics. Those components can be interpreted as:

:;<math>\color{Green}T_{12}</math>, <math>\color{Green}T_{13}</math>, <math>\color{Green}T_{23}</math>, <math>\color{Green}T_{21}</math>, <math>\color{Green}T_{31}</math>, <math>\color{Green}T_{32}</math>
::the components of ''shear stress,'' or stress applied tangential to the region

:;<math>\color{Orange}T_{11}</math>, <math>\color{Orange}T_{22}</math>, <math>\color{Orange}T_{33}</math>
::the components of ''normal stress,'' or stress applied perpendicular to the region; normal stress is another term for ''pressure.''

Pay particular attention to the first column of the above matrix: the components <math>\color{Purple}T_{00}</math>, <math>\color{Red}T_{10}</math>, <math>\color{Red}T_{20}</math> and <math>\color{Red}T_{30}</math>, are interpreted as ''densities.'' A density is what you get when you measure the flux of 4-momentum across a 3-surface of constant ''time.'' Put another way, the instantaneous value of 4-momentum flux is density.

Similarly, the diagonal space components of the stress-energy tensor — <math>\color{Orange}T_{11}</math>, <math>\color{Orange}T_{22}</math> and <math>\color{Orange}T_{33}</math> — represent normal stress, or ''pressure.'' Not some weird, relativistic pressure, but plain old ordinary ''pressure,'' like what keeps a balloon inflated. Pressure also contributes to gravitation, which raises a very interesting observation.

Imagine a box of air, a rigid box that won't flex. Let's say that the pressure of the air inside the box is the same as the pressure of the air outside the box. If we heat the box — assuming of course that the box is airtight — then the temperature of the gas inside will rise. In turn, as predicted by the [[ideal gas law]], the pressure within the box will increase.

''The box is now heavier than it was.''

More precisely, increasing the pressure inside the box raised the value of the pressure contribution to the stress-energy tensor, which will increase the curvature of spacetime around the box. What's more, merely increasing the ''temperature alone'' caused spacetime around the box to curve more, because the kinetic energy of the gas molecules inside the box also contributes to the stress-energy tensor, via the time-time component <math>\color{Purple}T_{00}</math>. ''All'' of these things contribute to the curvature of spacetime around the box, and thus to the gravitational field created by the box.

Of course, in practice, the contributions of increased pressure and kinetic energy would be miniscule compared to the mass contribution, so it would be extremely difficult to measure the gravitational effect of heating the box. But on larger scales, such as the [[sun]], pressure and temperature contribute significantly 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====

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

:<math>T_{\mu \nu }\ =\ 0</math>

This tensor represents a region of space in which there is no matter, energy or fields, not just at a given instant, but over the entire period of time in which we're interested in the region. Nothing exists in this region, and nothing happens in this region.

So one might assume that in a region where the stress-energy tensor is zero, the gravitational field must also necessarily be zero. There's nothing there to gravitate, so it follows naturally that there can be no gravitation.

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====

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.

;time-dependent
:The distribution of particles in our dust is not a constant; that is to say, the particles may be motion. The overall configuration you see when you look at the dust depends on the time at which you look at it, so the dust is said to be ''time-dependent.''

;identical
:The particles that make up our dust are all exactly the same; they don't differ from each other in any way.

;massive
:Each particle in our dust has some rest mass. Because the particles are all identical, their rest masses must also be identical. We'll call the rest mass of an individual particle <math>m_0</math>.

;non-interacting
:The particles don't interact with each other in any way: they don't collide, and they don't attract or repel each other. This is, of course, an idealization; since the particles are said to have mass <math>m_0</math>, they must ''at least'' interact with each other gravitationally, if not in other ways. But we're constructing our model in such a way that gravitational effects between the individual particles are so small as to be be negligible. Either the individual particles are very tiny, or the average distance between them is very large. This same assumption neatly cancels out any other possible interactions, as long as we assume that the particles are far enough apart.

;electrically neutral
:In addition to the obvious electrostatic effect of two charged particles either attracting or repelling each other — thus violating our "non-interacting" assumption — allowing the particles to be both charged and in motion would introduce electrodynamic effects that would have to be factored into the stress-energy tensor. We would ''greatly'' prefer to ignore these effects for the sake of simplicity, so by definition, the particles in our dust are all electrically neutral.

The easiest way to visualize an ideal dust is to imagine, well, dust. Dust particles sometimes catch the light of the sun and can be seen if you look closely enough. Each particle is moving in apparent ignorance of the rest, its velocity at any given moment dependent only on the motion of the air around it. If we take away the air, each particle of dust will continue moving in a straight line at a constant velocity, whatever its velocity happened to be at the time. This is a good visualization of an ideal dust.

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=====

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:

:<math>
\rho = m_0 n\,
</math>

where <math>m_0</math> is the mass of each particle and <math>n</math> is the number of particles in a cubical volume one unit of length on a side centered on <math>P</math>. This quantity is called ''proper density,'' meaning the density of the dust ''as measured within the dust's own reference frame.'' In other words, if we could somehow imagine the dust to measure ''its own density,'' the proper density is the number it would get.

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=====

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.

Directly measuring 4-velocity is an inherently tricky business, since one of its components describes motion along a "direction" that we cannot see with our eyes: motion through time. The math of [[special theory of relativity|special relativity]] lets us calculate the 4-velocity of a moving particle given only its 3-velocity <math>v</math> (with components <math>v^i</math> where <math>i=1, 2, 3</math>) and the speed of light. The time component of 4-velocity is given by:

:<math>
u^0 = \gamma c\,
</math>

and the space components <math>u^1</math>, <math>u^2</math> and <math>u^3</math> by:

:<math>
u^i = \gamma v^i\,
</math>

where <math>\gamma</math> is the ''boost,'' or Lorentz factor:

:<math>
\gamma = \frac{1}{\sqrt{1-\frac{\|v\|^2}{c^2}}}\,
</math>

and where <math>\|v\|^2</math>, in turn, is the square of the Euclidean magnitude of the 3-velocity vector <math>v</math>:

:<math>
\|v\|^2 = (v^1)^2+(v^2)^2+(v^3)^2\,
</math>

Therefore, if we know the 3-velocity of the dust at event <math>P</math>, then we can calculate its 4-velocity. (For more details on the how and why of 4-velocity, refer to the article on [[special theory of relativity|special relativity]].)

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=====

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

:<math>
T(x) = \rho (x) u(x)\otimes u(x)\,
</math>

where the symbol <math>\otimes</math> indicates a ''tensor product.'' The tensor product of two vectors is a tensor of rank two, so the stress-energy tensor must be a tensor of rank two. In an arbitrary coordinate frame <math>x^\mu</math>, the contravariant components of the stress-energy tensor for an ideal dust are given by:

:<math>
T^{\mu\nu} = \rho u^{\mu} u^{\nu}\,
</math>

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

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

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

:<math>
T^{00} = \rho u^0 u^0 = \rho (\gamma c) (\gamma c) = \rho \gamma^2 c^2 \,
</math>

If we rearrange the terms in this equation slightly, something important becomes apparent:

:<math>
T^{00} = \gamma^2 (\rho c^2)\,
</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>

======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:

:<math>
T^{10} = \rho u^1 u^0 = \rho (\gamma v^1) (\gamma c) = \gamma^2 c \rho v^1\,
</math>

Again, recall that <math>\rho</math> is a quantity of mass per unit volume. Multiplying a mass times a velocity gives ''momentum,'' so we can interpret <math>\rho v^1</math> as the ''density of momentum'' along the <math>x^1</math> direction, multiplied by constants <math>c</math> and <math>\gamma^2</math>. Momentum density is an extremely difficult quantity to visualize, but it's a quantity that comes up over and over in general relativity. If nothing else, one can take comfort in the fact that momentum density is mathematically equivalent to the product of mass density and velocity, both of which are much more intuitive quantities.

Note that the off-diagonal components of the tensor are equal to each other:

:<math>
T^{10} = \rho u^1 u^0 = \rho (\gamma v^1) (\gamma c) = \rho \gamma^2 c v^1\,
</math>

:<math>
T^{01} = \rho u^0 u^1 = \rho (\gamma c) (\gamma v^1) = \rho c \gamma^2 v^1\,
</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======

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

:<math>
T^{11} = \rho u^1 u^1 = \rho (\gamma v^1) (\gamma v^1) = \gamma^2 \rho (v^1)^2\,
</math>

In this case, we're multiplying a four-dimensional mass density, <math>\rho</math>, by the square of a component of 4-velocity. By [[dimensional analysis]], we can see:

:<math>
\frac{\mbox{kg}}{\mbox{m}^4} \cdot \frac{\mbox{m}^2}{\mbox{s}^2} = \frac{\mbox{kg} \cdot \mbox{m}^2}{\mbox{m}^4 \cdot \mbox{s}^2} = \frac{\mbox{kg}}{\mbox{m}^2 \cdot \mbox{s}^2}
</math>

Recall that the [[force]] has units:

:<math>
\frac{\mbox{kg} \cdot \mbox{m}}{\mbox{s}^2}
</math>

If we divide the units of the diagonal space component by the units of force, we get:

:<math>
\frac{\mbox{kg}}{\mbox{m}^2 \cdot \mbox{s}^2} \cdot \frac{\mbox{s}^2}{\mbox{kg} \cdot \mbox{m}} = \frac{1}{\mbox{m}^3}
</math>

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======

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>

:<math>T_{\mu \nu }=\begin{pmatrix}

\gamma^2 \rho c^2 & \gamma^2 \rho v^1 c & \gamma^2 \rho v^2 c & \gamma^2 \rho v^3 c \\
\gamma^2 \rho c v^1 & \gamma^2 \rho (v^1)^2 & \gamma^2 \rho v^2 v^1 & \gamma^2 \rho v^3 v^1 \\
\gamma^2 \rho c v^2 & \gamma^2 \rho v^1 v^2 & \gamma^2 \rho (v^2)^2 & \gamma^2 \rho v^3 v^2 \\
\gamma^2 \rho c v^3 & \gamma^2 \rho v^1 v^3 & \gamma^2 \rho v^2 v^3 & \gamma^2 \rho (v^3)^2

\end{pmatrix}</math>

The large-scale structure of the tensor now becomes apparent. This is the stress-energy tensor of an ideal dust. The tensor is composed entirely out of the proper density and the components of 4-velocity. When velocities are low, the coefficient <math>\gamma^2</math>, even though it's a squared value, remains extremely close to one.

The time-time component includes a mass multiplied by the square of the speed of light, so it has to do with energy. The rest of the top row and left column all include the speed of light as a coefficient, as well as density and velocity; in the case of an ideal dust which is made up of non-interacting particles, the energy flux along any basis direction is the same as the momentum density along that direction. This is not the case in other, less simple models, but it's true here.

The diagonal space components of the tensor represent pressure. For example, the <math>T^{11}</math> component represents the pressure that would be exerted on a plane perpendicular to the <math>x^1</math> direction.

The off-diagonal space components represent shear stress. The <math>T^{12}</math> component, for instance, represents the pressure that would be exerted in the <math>x^2</math> direction on a plane perpendicular to the <math>x^1</math> axis.

The overall process for calculating the stress-energy tensor for any system is fairly similar to the example given here. It involves taking into account all the matter and energy in the system, describing how the system evolves over time, and breaking that evolution down into components which represent individual densities and fluxes along different directions relative to a chosen coordinate basis.

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===

==Exact Solutions in General Relativity==

==Tests of General Relativity==

General relativity provides one explanation for the precession of Mercury's perihelion, which was moving at a different speed than that predicted by a simple application of Newton's law of universal gravitation. (Previous scientists had attempted to explain it by the gravitational pull of a hypothetical planet inside Mercury's orbit, which they called [[Vulcan]]. This could also be explained by altering the precise inverse-square relation of Newtonian gravity to distance, but that was disfavored by mathematicians due to its inelegance in integrating.)

British historian Paul Johnson declares the turning point in the acceptance of general relativity to have been when Sir [[Arthur Eddington]], an esteemed English astronomer, ventured out on a boat off Africa in 1919 with a local Army unit to observe the bending of starlight around the sun during a total [[solar eclipse]]. General relativity predicted that the light from the stars would be bent due to passing close to the sun. (A smaller degree of bending could also be consistent with Newton's theory, if one hypothesized light to consist of particles. However, that [[particle theory of light]] had gone out of favor previously.) Eddington detected a bending of light, but his range of error overlapped both Einstein's and Newton's predictions. Upon his return to England, Eddington declared that his observations proven the theory of relativity. His experiment was later confirmed by more rigorous experiments, such as those performed by the [[Hubble Space Telescope]] <ref>[http://www.spaceimages.com/gravlen.html Hubble Gravitational Lens Photo]</ref><ref> [http://csep10.phys.utk.edu/astr162/lect/galaxies/lensing.html Gravitational Lensing] </ref><ref>[http://www.iam.ubc.ca/~newbury/lenses/glgallery.html]</ref>. Lorentz has this to say on the discrepancies between the empirical eclipse data and Einstein's predictions.

::''It indeed seems that the discrepancies may be ascribed to faults in observations, which supposition is supported by the fact that the observations at Prince's Island, which, it is true, did not turn out quite as well as those mentioned above, gave the result, of 1.64, somewhat lower than Einstein's figure.''<ref>Lorentz, H.A. [http://ia331314.us.archive.org/2/items/theeinsteintheor11335gut/11335-h/11335-h.htm The Einstein Theory of Relativity]</ref>

==Relation to Special Relativity==

[[Special relativity]] is the limiting case of general relativity where all gravitational fields are weak. Alternatively, special relativity is the limiting case of general relativity when all reference frames are inertial (non-accelerating and without gravity).

==External Links==
*Einstein, Albert (1916), [http://www.alberteinstein.info/gallery/pdf/CP6Doc30_English_pp146-200.pdf "The Foundation of the General Theory of Relativity"] (PDF), Annalen der Physik 49

==References==
{{reflist}}

[[Category:Physics]]

General theory of relativity

2009-11-15T02:46:13Z

2009-11-14T22:56:41Z

KSorenson: /* Action at a distance */

The 1300s did not occur after the [[King James Bible]], and hence my reversion. You might try taking [[World History Final Exam]].--[[User:Aschlafly|Andy Schlafly]] 16:16, 3 June 2009 (EDT)

:Oh, sorry. I didn't read the fine print "developed since the King James Version". That explains a bunch of conservative words that I found surprisingly absent, like "work" and "decency". I'll be more selective and careful in the future.

:My KJV doesn't show a date in the dedication, which sort of surprised me. But I knew it was during the reign of James I, in the early 1600's. I didn't know, until just now, that it was 1611.

:The Final Exam question you are referring to was 40, right? Answer is B of course. I'll look at the exam in more detail soon. I've just finished taking some other exams. [[User:SaraT|SaraT]] 17:47, 3 June 2009 (EDT)

Here are my answers to the World History Final Exam:

:1 c 11 b 21 b 31 d 41 d 51 c
:2 d 12 b 22 d 32 d 42 c 52 b
:3 a 13 a 23 b 33 b 43 b 53 a
:4 c 14 c 24 b 34 c 44 b 54 d
:5 d 15 e 25 b 35 d 45 e 55 b
:6 a 16 e 26 d 36 b 46 e 56 e
:7 b 17 d 27 b 37 a 47 d 57 d
:8 a 18 c 28 e 38 c 48 c 58 b
:9 c 19 b 29 c 39 e 49 b 59 c
:10 b 20 d 30 d 40 b 50 d 60 a
:EC d for boys, e for girls

Minor nit on question 53: It's Carta, not Charta.

MAJOR nit on question 59: Solving encrypted messages by checking letter frequencies is the stuff of puzzles in popular magazines and children's books. The Enigma cipher was '''much''' more difficult than that. The rotors advanced, completely changing the permutation, after every letter. Solving the Enigma cipher required the concentrated effort of the best minds of the British mathematics community. See "Alan Turing: The Enigma" by Andrew Hodges for a readable, but not ''too'' technical account, of the problem and the way it was attacked.

: I don't know if I'll bother grading your exam, because you defiantly answered both the boys and girls extra credit question. You're not both.

: Your "minor nit on question 53" is wrong. See [http://www.magnacharta.org/]

: Your "major nit on question 59" is also wrong. British mathematicians did not crack the Enigma; Polish mathematicians did, and part of the decrypting was based on letter frequency. Godspeed.--[[User:Aschlafly|Andy Schlafly]] 22:04, 4 June 2009 (EDT)

::"Your "minor nit on question 53" is wrong." Actually SaraT is correct. The Magna Carta is the Magna Carta, or Magna Carta Libertatum. The translation of Carta is Charter, but any spelling of Carta as Charta is completely incorrect. I'm afraid to say that the people running the site that you linked too need to educate themselves re:13th century Latin spelling.

:::"Your "major nit on question 59" is also wrong. British mathematicians did not crack the Enigma; Polish mathematicians did, and part of the decrypting was based on letter frequency. Godspeed." Again incorrect. Whilst the Polish did crack the pre-war Enigma code the German's changed the way that Enigma generated messages specifically to counter the way that the Polish had cracked the pre-war code. It was the British who cracked the new Enigma code during the war. Because of the way that encryption using the Enigma machine was changed, plus operational changes such as not allowing messages of more than 250 words, this meant the process that you refer to in the question, initialization vector which is the system that the Biuro Szyfrów used to crack the pre-war code, became almost useless in cracking the new Enigma code.--[[User:DanHutchin|DanHutchin]] 17:23, 14 November 2009 (EST)

== Action at a distance ==

SaraT and KSorenson (this is being sent to both):

I'm interested in working on the article about [[action at a distance]]. (Actually I'm busy working on many things about science and math, as a look at my [[Special:Contributions/PatrickD|contributions]] will show, but this particular issue has bubbled to the top of my agenda.) I'm confused about the ways this term is used. Since you both seem to be interested in the history of science, you might be able to shed some light on it.

The suggested meanings I have come across are:
#Exertion of force on an object that isn't in direct contact. This must be what it means in Newtonian gravity, right?
#Transmission of a force instantaneously, or, in any case, supraluminally. This doesn't seem to have been an issue with Newton, because the finite speed of light, and the causality consequences of that, weren't known at the time. Right?
#Consequences of "Quantum Entanglement". This seems to relate to supraluminal transmission of information, and is therefore related to #2. Right?

So which is it? Can either of you enlighten me? Or fix the article? (By the way, you both write very well.) [[User:PatrickD|PatrickD]] 22:28, 12 November 2009 (EST)

I've seen what KSorenson wrote, and her explanation of the locality and causality aspects are far superior to anything I could do. But I would like to point out that, historically, it meant something different. One thing that the old and new definitions have in common is an element of "spookiness". Of course, what was spooky in the 1600's is perfectly sensible now.

The term is considered rather quaint now; I think it should properly be used only in the old context. As KS points out, in the modern world of field theories, "locality" and "non-locality" are better terms. If something (like an electric charge) locally affects the field (electric field or Faraday tensor) here, and there are equations (Maxwell's equations) relating the behavior of the field here with the behavior there, and the field there causes a particle there to feel a force, that is not really "action at a distance". It's perfectly acceptable. In modern terms, it's a local field theory. The field "mediated" the force. The way the equations relate the field at one point to the field at another point (divergences and such) that carry the force from "here" to "there".

But the carrying of force from one point to another wasn't so easy to accept in the 1600's. It was considered spooky, just as Schrodinger's cat and quantum entanglement are considered spooky today. In the 1600's, people generally believed that one object could only exert a force on another object if the two were in physical contact. There was plenty of practical experience underlying this belief.

:But there were two effects that I can think of that violated this: There were magnets--natural lodestones; and their ability to point North had been known since antiquity. And there was gravity. The apple fell to the ground without anyone pushing it. How did they reconcile this? My guess is that they just didn't think as hard about such issues as perhaps they might have.

I think that Newton's discovery that gravity makes the planets move forced people to think about what they had been ignoring. You can overlook an apple falling to the ground, but you can't overlook a force making Mars move. Was it the fact that the force operated through a vacuum that made people uneasy? Or was it the fact that the distances were so vast? I suspect it was a little of each. That which had been overlooked before could no longer be denied. There was action at a distance. Across hundreds of millions of miles of empty space. This bothered people. It even bothered Newton. He wrote about not believing in action at a distance, even though he had just devised the world's first field theory. (He was not the only person to have a hard time coming to grips with his own discoveries.)

Action at a distance has also been used, classically, to refer to action traveling faster than the speed of light. But this did not come up in Newton's time. That there should be a finite speed of propagation wasn't an issue. In fact, Newton proved that angular momentum would not be preserved unless gravity propagated instantaneously.

:How do we reconcile that with the fact that it is propagated at the finite speed of light? Classical mechanics is completely consistent with special relativity under the assumption that the speed of light is infinite. Newton was doing a relativistically correct proof, as long as c = infinity.

Later on, as the finiteness of the speed of light became known, the questions arose whether the effects of fields (electric, magnetic, and gravitational) could travel faster than the speed of light. Such an effect might have been called "action at a distance", and would have been spooky. But the issue was pretty much settled in the 19th and early 20th centuries.

The issue arose again in the mid 20th century, referring to the possibility that fields (this time the quantum field) might mediate actions faster than light. The problem is often posited in terms of transmission of information faster than light, which would violate common assumptions about causality.

And that's where we stand now, though "non-locality" is a much better modern term. I hope this helps.

[[User:SaraT|SaraT]] 14:44, 14 November 2009 (EST)

:I've heard, rarely but occasionally, the phrase "action at a distance" to refer to one particle acting on another particle that's outside its light cone. Have either of you heard that one? Maybe we should include that interpretation of the phrase as well? --[[User:KSorenson|KSorenson]] 17:56, 14 November 2009 (EST)

Talk:Essay:Quantifying Order

2009-11-14T22:48:26Z

KSorenson: Can you please cite the "55 arc seconds per century" bit?

General theory of relativity

2009-11-14T21:30:58Z

KSorenson: moved math-ahead template to where it belongs

{{Progress}}

The '''general theory of relativity''' is a theory of gravity that is compatible with [[Special theory of relativity|Special Relativity]]. The theory was first published by [[Albert Einstein]] in 1916.

The general theory of relativity is a ''metric theory,'' sometimes also called a ''geometric theory.'' Metric theories describe physical phenomena in terms of [[differential geometry]]. This stands in contrast to Isaac Newton's [[Law of Universal Gravitation|law of universal gravitation]], which described gravity in terms of a [[vector field]]. In the case of general relativity, the theory relates ''stress-energy'' — an extension of the concept of [[mass]] — and the [[curvature]] of [[spacetime]]. In the words of physicist John Wheeler, "Space tells matter how to move, matter tells space how to curve."<ref>Misner, Thorne & Wheeler. ''Gravitation.'' (1973)</ref>

In the [[weak field approximation]], where velocities of moving objects are low and gravitational fields are not very severe, the theory of general relativity is said to ''reduce to'' the law of universal gravitation. That is to say, under those circumstances the equations of general relativity are mathematically equivalent to the equations of Newtonian gravitation.

The theory was inspired by a thought experiment developed by Einstein involving two elevators. The first elevator is stationary on the Earth, while the other is being pulled through space at a constant acceleration of g. Einstein realized that any physical experiment carried out in the elevators would give the same result. This realization is known as the equivalence principle and it states that accelerating frames of reference and gravitational fields are indistinguishable. General Relativity is the theory of gravity that incorporates Special Relativity and the equivalence principle.

==Qualitative Introduction to General Relativity==

The relationship between the curvature of spacetime and the motions of freely falling bodies is often explained by an easily imagined analogy: bowling balls and golf balls on a trampoline.

Imagine that we place a golf ball on an ordinary backyard trampoline. If we give the golf ball a slight push, it will roll along in a straight line until friction brings it to a halt. But if we imagine that friction doesn't exist, then the golf ball will roll in a straight line at a constant speed forever — or at least until it reaches the edge of the trampoline and falls off.

Now imagine a bowling ball sitting in the middle of a trampoline. The trampoline isn't a rigid surface, so it deforms where the weight of the bowling ball pushes it down. This causes the surface to be curved downward, toward the ground.

If we place a golf ball near the edge of the trampoline, it will begin to roll toward the bowling ball, because the trampoline is sloped downward in that direction. The golf ball will start off moving very slowly, then pick up speed as it approaches the bowling ball, until finally it collides with the bowling ball and comes to rest.

But if we give the golf ball a slight push in a direction perpendicular to the direction of the bowling ball, then it will move in a curved path. If we only push it a little bit, the golf ball will curve slightly, but still collide with the bowling ball. If we push the golf ball somewhat harder, it will curve toward the bowling ball, pass by it on one side and climb back out of the depression until it reaches the edge and falls off.

But if we're ''very'' careful, and give the golf ball ''just the right'' push, it will curve completely around the bowling ball and return to our hand.

This is, in a nutshell, how spacetime and matter interact under the theory of general relativity. Massive objects — represented in our analogy by the bowling ball — curve spacetime. Less-massive objects also curve spacetime, but to a lesser extent. If the object is small enough, like our golf ball, the amount of curvature is so slight that we can't even measure it.

The way the golf ball moved in the three scenarios we imagined correspond to ''conic-section orbits,'' or ''Kepler orbits.'' When we just placed the golf ball and it rolled straight toward the bowling ball, that was a ''degenerate'' orbit: a straight line. When we gave it a push and it curved around the bowling ball and off the edge of the trampoline, that was a ''hyperbolic'' orbit. And when we gave it just the right push so that it curved around the bowling ball and back to our hand, that was an ''elliptical'' orbit.

These are the same orbits that are predicted by Isaac Newton's [[Law of Universal Gravitation|law of universal gravitation]]. But in Newton's equations, objects move in conic-section orbits because of a force that accelerates them toward the central mass. In general relativity, objects move in conic-section orbits because spacetime itself is curved, just like our imaginary trampoline was curved by the bowling ball.

Of course, our analogy is far from perfect. Our imaginary trampoline curved ''downward,'' toward the ground, pushed down by the weight of the bowling ball. That's not how spacetime behaves in general relativity. It curves, but not ''toward'' anything, not in any ''direction.'' Spacetime in general relativity is instead said to have ''intrinsic curvature,'' which is mathematically quite simple but very difficult to visualize.

And of course there are many, many other aspects of general relativity that our imaginary trampoline didn't model. But the analogy captures the essential nature of the theory: the bowling ball caused the trampoline to be curved, and the curvature of the trampoline caused the golf ball to move in a different way than if the trampoline had been flat. This is the essence of general relativity: matter tells space how to curve, and space tells matter how to move.

==Quantitative Introduction to General Relativity==

{{Template: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.

The mathematical expression of the theory of general relativity takes the form of the ''Einstein field equations,'' a set of ten nonlinear partial differential equations. While solving these equations is quite difficult, examining them provides valuable insight into the structure and meaning of the theory.

In their general form, the Einstein field equations are written as a single [[tensor]] equation in [[abstract index notation]] relating the curvature of spacetime to sources of curvature such as energy density and momentum.

:<math> G_{\mu\nu} = 8\,\pi\,G\,T_{\mu\nu} </math>

In this form, <math>G_{\mu\nu}</math> represents the ''Einstein tensor,'' <math>G</math> is the same ''gravitational constant'' that appears in the [[Law of Universal Gravitation|law of universal gravitation]], and <math>T_{\mu\nu}</math> is the ''stress-energy tensor'' (sometimes referred to as the ''energy-momentum tensor).'' The indices <math>\mu</math> and <math>\nu</math> range from zero to three, representing the time coordinate and the three space coordinates in a manner consistent with [[Special theory of relativity|special relativity]].

The left side of the equation — the Einstein tensor — describes the curvature of spacetime in the region under examination. The right side of the equation describes everything in that region that affects the curvature of spacetime.

As we can clearly see even in this simplified form, the Einstein field equations can be solved "in either direction." Given a description of the gravitating matter, energy, momentum and fields in a region of spacetime, we can calculate the curvature of spacetime surrounding that region. On the other hand, given a description of the curvature of a region spacetime, we can calculate the motion of a test particle anywhere within that region.

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===

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.

First we will define the stress-energy tensor technically, then we'll examine what that definition means. In technical terms, the stress energy tensor represents ''the flux of the <math>\mu</math> component of 4-momentum across a surface of constant coordinate <math>x^\nu</math>.''

Fine. But what does that ''mean?''

In classical mechanics, it's customary to refer to coordinates in space as <math>x</math>, <math>y</math> and <math>z</math>. In general relativity, the convention is to talk instead about coordinates <math>x^0</math>, <math>x^1</math>, <math>x^2</math>, and <math>x^3</math>, where <math>x^0</math> is the time coordinate otherwise called <math>t</math>, and the other three are just the <math>x</math>, <math>y</math> and <math>z</math> coordinates. So "a surface of constant coordinate "<math>x^\nu</math>" simply means a 3-plane perpendicular to the <math>x^\nu</math> axis.

The ''flux'' of a quantity can be visualized as the magnitude of the current in a river: the flux of water is the amount of water that passes through a cross-section of the river in a given interval of time. So more generally, the flux of a quantity across a surface is the amount of that quantity that passes through that surface.

''Four-momentum'' is the special relativity analogue of the familiar momentum from classical mechanics, with the property that the time coordinate <math>\mathbf{P}^0</math> of a particle's four-momentum is simply the energy of the particle; the other three components of four-momentum are the same as in classical momentum.

So putting that all together, the stress-energy tensor is the flux of 4-momentum across a surface of constant coordinate. In other words, the stress-energy tensor describes the density of energy and momentum, and the flux of energy and momentum in a region. Since under the [[mass-energy equivalence]] principle we can convert mass units to energy units and vice-versa, this means that the stress-energy tensor describes ''all the mass and energy in a given region of spacetime.''

Put even more simply, the stress-energy tensor represents ''everything that gravitates.''

The stress-energy tensor, being a tensor of rank two in four-dimensional spacetime, has sixteen components that can be written as a 4 × 4 matrix.

:<math>T_{\mu \nu }=\begin{pmatrix}
\color{Purple}T_{00} & \color{Blue}T_{01} & \color{Blue}T_{02} & \color{Blue}T_{03} \\
\color{Red}T_{10} & \color{Orange}T_{11} & \color{Green}T_{12} & \color{Green}T_{13} \\
\color{Red}T_{20} & \color{Green}T_{21} & \color{Orange}T_{22} & \color{Green}T_{23} \\
\color{Red}T_{30} & \color{Green}T_{31} & \color{Green}T_{32} & \color{Orange}T_{33}
\end{pmatrix}</math>

Here the components have been color-coded to help clarify their physical interpretations.

:;<math>\color{Purple}T_{00}</math>
::energy density, which is equivalent to mass-energy density; this component includes the mass contribution

:;<math>\color{Red}T_{10}</math>, <math>\color{Red}T_{20}</math>, <math>\color{Red}T_{30}</math>
::the components of momentum density

:;<math>\color{Blue}T_{01}</math>, <math>\color{Blue}T_{02}</math>, <math>\color{Blue}T_{03}</math>
::the components of energy flux

The space-space components of the stress-energy tensor are simply the [[stress tensor]] from classic mechanics. Those components can be interpreted as:

:;<math>\color{Green}T_{12}</math>, <math>\color{Green}T_{13}</math>, <math>\color{Green}T_{23}</math>, <math>\color{Green}T_{21}</math>, <math>\color{Green}T_{31}</math>, <math>\color{Green}T_{32}</math>
::the components of ''shear stress,'' or stress applied tangential to the region

:;<math>\color{Orange}T_{11}</math>, <math>\color{Orange}T_{22}</math>, <math>\color{Orange}T_{33}</math>
::the components of ''normal stress,'' or stress applied perpendicular to the region; normal stress is another term for ''pressure.''

Pay particular attention to the first column of the above matrix: the components <math>\color{Purple}T_{00}</math>, <math>\color{Red}T_{10}</math>, <math>\color{Red}T_{20}</math> and <math>\color{Red}T_{30}</math>, are interpreted as ''densities.'' A density is what you get when you measure the flux of 4-momentum across a 3-surface of constant ''time.'' Put another way, the instantaneous value of 4-momentum flux is density.

Similarly, the diagonal space components of the stress-energy tensor — <math>\color{Orange}T_{11}</math>, <math>\color{Orange}T_{22}</math> and <math>\color{Orange}T_{33}</math> — represent normal stress, or ''pressure.'' Not some weird, relativistic pressure, but plain old ordinary ''pressure,'' like what keeps a balloon inflated. Pressure also contributes to gravitation, which raises a very interesting observation.

Imagine a box of air, a rigid box that won't flex. Let's say that the pressure of the air inside the box is the same as the pressure of the air outside the box. If we heat the box — assuming of course that the box is airtight — then the temperature of the gas inside will rise. In turn, as predicted by the [[ideal gas law]], the pressure within the box will increase.

''The box is now heavier than it was.''

More precisely, increasing the pressure inside the box raised the value of the pressure contribution to the stress-energy tensor, which will increase the curvature of spacetime around the box. What's more, merely increasing the ''temperature alone'' caused spacetime around the box to curve more, because the kinetic energy of the gas molecules inside the box also contributes to the stress-energy tensor, via the time-time component <math>\color{Purple}T_{00}</math>. ''All'' of these things contribute to the curvature of spacetime around the box, and thus to the gravitational field created by the box.

Of course, in practice, the contributions of increased pressure and kinetic energy would be miniscule compared to the mass contribution, so it would be extremely difficult to measure the gravitational effect of heating the box. But on larger scales, such as the [[sun]], pressure and temperature contribute significantly 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====

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

:<math>T_{\mu \nu }\ =\ 0</math>

This tensor represents a region of space in which there is no matter, energy or fields, not just at a given instant, but over the entire period of time in which we're interested in the region. Nothing exists in this region, and nothing happens in this region.

So one might assume that in a region where the stress-energy tensor is zero, the gravitational field must also necessarily be zero. There's nothing there to gravitate, so it follows naturally that there can be no gravitation.

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====

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.

;time-dependent
:The distribution of particles in our dust is not a constant; that is to say, the particles may be motion. The overall configuration you see when you look at the dust depends on the time at which you look at it, so the dust is said to be ''time-dependent.''

;identical
:The particles that make up our dust are all exactly the same; they don't differ from each other in any way.

;massive
:Each particle in our dust has some rest mass. Because the particles are all identical, their rest masses must also be identical. We'll call the rest mass of an individual particle <math>m_0</math>.

;non-interacting
:The particles don't interact with each other in any way: they don't collide, and they don't attract or repel each other. This is, of course, an idealization; since the particles are said to have mass <math>m_0</math>, they must ''at least'' interact with each other gravitationally, if not in other ways. But we're constructing our model in such a way that gravitational effects between the individual particles are so small as to be be negligible. Either the individual particles are very tiny, or the average distance between them is very large. This same assumption neatly cancels out any other possible interactions, as long as we assume that the particles are far enough apart.

;electrically neutral
:In addition to the obvious electrostatic effect of two charged particles either attracting or repelling each other — thus violating our "non-interacting" assumption — allowing the particles to be both charged and in motion would introduce electrodynamic effects that would have to be factored into the stress-energy tensor. We would ''greatly'' prefer to ignore these effects for the sake of simplicity, so by definition, the particles in our dust are all electrically neutral.

The easiest way to visualize an ideal dust is to imagine, well, dust. Dust particles sometimes catch the light of the sun and can be seen if you look closely enough. Each particle is moving in apparent ignorance of the rest, its velocity at any given moment dependent only on the motion of the air around it. If we take away the air, each particle of dust will continue moving in a straight line at a constant velocity, whatever its velocity happened to be at the time. This is a good visualization of an ideal dust.

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=====

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:

:<math>
\rho = m_0 n\,
</math>

where <math>m_0</math> is the mass of each particle and <math>n</math> is the number of particles in a cubical volume one unit of length on a side centered on <math>P</math>. This quantity is called ''proper density,'' meaning the density of the dust ''as measured within the dust's own reference frame.'' In other words, if we could somehow imagine the dust to measure ''its own density,'' the proper density is the number it would get.

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=====

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.

Directly measuring 4-velocity is an inherently tricky business, since one of its components describes motion along a "direction" that we cannot see with our eyes: motion through time. The math of [[special theory of relativity|special relativity]] lets us calculate the 4-velocity of a moving particle given only its 3-velocity <math>v</math> (with components <math>v^i</math> where <math>i=1, 2, 3</math>) and the speed of light. The time component of 4-velocity is given by:

:<math>
u^0 = \gamma c\,
</math>

and the space components <math>u^1</math>, <math>u^2</math> and <math>u^3</math> by:

:<math>
u^i = \gamma v^i\,
</math>

where <math>\gamma</math> is the ''boost,'' or Lorentz factor:

:<math>
\gamma = \frac{1}{\sqrt{1-\frac{\|v\|^2}{c^2}}}\,
</math>

and where <math>\|v\|^2</math>, in turn, is the square of the Euclidean magnitude of the 3-velocity vector <math>v</math>:

:<math>
\|v\|^2 = (v^1)^2+(v^2)^2+(v^3)^2\,
</math>

Therefore, if we know the 3-velocity of the dust at event <math>P</math>, then we can calculate its 4-velocity. (For more details on the how and why of 4-velocity, refer to the article on [[special theory of relativity|special relativity]].)

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=====

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

:<math>
T(x) = \rho (x) u(x)\otimes u(x)\,
</math>

where the symbol <math>\otimes</math> indicates a ''tensor product.'' The tensor product of two vectors is a tensor of rank two, so the stress-energy tensor must be a tensor of rank two. In an arbitrary coordinate frame <math>x^\mu</math>, the contravariant components of the stress-energy tensor for an ideal dust are given by:

:<math>
T^{\mu\nu} = \rho u^{\mu} u^{\nu}\,
</math>

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

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

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

:<math>
T^{00} = \rho u^0 u^0 = \rho (\gamma c) (\gamma c) = \rho \gamma^2 c^2 \,
</math>

If we rearrange the terms in this equation slightly, something important becomes apparent:

:<math>
T^{00} = \gamma^2 (\rho c^2)\,
</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>

======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:

:<math>
T^{10} = \rho u^1 u^0 = \rho (\gamma v^1) (\gamma c) = \gamma^2 c \rho v^1\,
</math>

Again, recall that <math>\rho</math> is a quantity of mass per unit volume. Multiplying a mass times a velocity gives ''momentum,'' so we can interpret <math>\rho v^1</math> as the ''density of momentum'' along the <math>x^1</math> direction, multiplied by constants <math>c</math> and <math>\gamma^2</math>. Momentum density is an extremely difficult quantity to visualize, but it's a quantity that comes up over and over in general relativity. If nothing else, one can take comfort in the fact that momentum density is mathematically equivalent to the product of mass density and velocity, both of which are much more intuitive quantities.

Note that the off-diagonal components of the tensor are equal to each other:

:<math>
T^{10} = \rho u^1 u^0 = \rho (\gamma v^1) (\gamma c) = \rho \gamma^2 c v^1\,
</math>

:<math>
T^{01} = \rho u^0 u^1 = \rho (\gamma c) (\gamma v^1) = \rho c \gamma^2 v^1\,
</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======

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

:<math>
T^{11} = \rho u^1 u^1 = \rho (\gamma v^1) (\gamma v^1) = \gamma^2 \rho (v^1)^2\,
</math>

In this case, we're multiplying a four-dimensional mass density, <math>\rho</math>, by the square of a component of 4-velocity. By [[dimensional analysis]], we can see:

:<math>
\frac{\mbox{kg}}{\mbox{m}^4} \cdot \frac{\mbox{m}^2}{\mbox{s}^2} = \frac{\mbox{kg} \cdot \mbox{m}^2}{\mbox{m}^4 \cdot \mbox{s}^2} = \frac{\mbox{kg}}{\mbox{m}^2 \cdot \mbox{s}^2}
</math>

Recall that the [[force]] has units:

:<math>
\frac{\mbox{kg} \cdot \mbox{m}}{\mbox{s}^2}
</math>

If we divide the units of the diagonal space component by the units of force, we get:

:<math>
\frac{\mbox{kg}}{\mbox{m}^2 \cdot \mbox{s}^2} \cdot \frac{\mbox{s}^2}{\mbox{kg} \cdot \mbox{m}} = \frac{1}{\mbox{m}^3}
</math>

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======

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>

:<math>T_{\mu \nu }=\begin{pmatrix}

\gamma^2 \rho c^2 & \gamma^2 \rho v^1 c & \gamma^2 \rho v^2 c & \gamma^2 \rho v^3 c \\
\gamma^2 \rho c v^1 & \gamma^2 \rho (v^1)^2 & \gamma^2 \rho v^2 v^1 & \gamma^2 \rho v^3 v^1 \\
\gamma^2 \rho c v^2 & \gamma^2 \rho v^1 v^2 & \gamma^2 \rho (v^2)^2 & \gamma^2 \rho v^3 v^2 \\
\gamma^2 \rho c v^3 & \gamma^2 \rho v^1 v^3 & \gamma^2 \rho v^2 v^3 & \gamma^2 \rho (v^3)^2

\end{pmatrix}</math>

The large-scale structure of the tensor now becomes apparent. This is the stress-energy tensor of an ideal dust. The tensor is composed entirely out of the proper density and the components of 4-velocity. When velocities are low, the coefficient <math>\gamma^2</math>, even though it's a squared value, remains extremely close to one.

The time-time component includes a mass multiplied by the square of the speed of light, so it has to do with energy. The rest of the top row and left column all include the speed of light as a coefficient, as well as density and velocity; in the case of an ideal dust which is made up of non-interacting particles, the energy flux along any basis direction is the same as the momentum density along that direction. This is not the case in other, less simple models, but it's true here.

The diagonal space components of the tensor represent pressure. For example, the <math>T^{11}</math> component represents the pressure that would be exerted on a plane perpendicular to the <math>x^1</math> direction.

The off-diagonal space components represent shear stress. The <math>T^{12}</math> component, for instance, represents the pressure that would be exerted in the <math>x^2</math> direction on a plane perpendicular to the <math>x^1</math> axis.

The overall process for calculating the stress-energy tensor for any system is fairly similar to the example given here. It involves taking into account all the matter and energy in the system, describing how the system evolves over time, and breaking that evolution down into components which represent individual densities and fluxes along different directions relative to a chosen coordinate basis.

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===

==Exact Solutions in General Relativity==

==Tests of General Relativity==

General relativity provides one explanation for the precession of Mercury's perihelion, which was moving at a different speed than that predicted by a simple application of Newton's law of universal gravitation. (Previous scientists had attempted to explain it by the gravitational pull of a hypothetical planet inside Mercury's orbit, which they called [[Vulcan]]. This could also be explained by altering the precise inverse-square relation of Newtonian gravity to distance, but that was disfavored by mathematicians due to its inelegance in integrating.)

British historian Paul Johnson declares the turning point in the acceptance of general relativity to have been when Sir [[Arthur Eddington]], an esteemed English astronomer, ventured out on a boat off Africa in 1919 with a local Army unit to observe the bending of starlight around the sun during a total [[solar eclipse]]. General relativity predicted that the light from the stars would be bent due to passing close to the sun. (A smaller degree of bending could also be consistent with Newton's theory, if one hypothesized light to consist of particles. However, that [[particle theory of light]] had gone out of favor previously.) Eddington detected a bending of light, but his range of error overlapped both Einstein's and Newton's predictions. Upon his return to England, Eddington declared that his observations proven the theory of relativity. His experiment was later confirmed by more rigorous experiments, such as those performed by the [[Hubble Space Telescope]] <ref>[http://www.spaceimages.com/gravlen.html Hubble Gravitational Lens Photo]</ref><ref> [http://csep10.phys.utk.edu/astr162/lect/galaxies/lensing.html Gravitational Lensing] </ref><ref>[http://www.iam.ubc.ca/~newbury/lenses/glgallery.html]</ref>. Lorentz has this to say on the discrepancies between the empirical eclipse data and Einstein's predictions.

::''It indeed seems that the discrepancies may be ascribed to faults in observations, which supposition is supported by the fact that the observations at Prince's Island, which, it is true, did not turn out quite as well as those mentioned above, gave the result, of 1.64, somewhat lower than Einstein's figure.''<ref>Lorentz, H.A. [http://ia331314.us.archive.org/2/items/theeinsteintheor11335gut/11335-h/11335-h.htm The Einstein Theory of Relativity]</ref>

==Relation to Special Relativity==

[[Special relativity]] is the limiting case of general relativity where all gravitational fields are weak. Alternatively, special relativity is the limiting case of general relativity when all reference frames are inertial (non-accelerating and without gravity).

==External Links==
*Einstein, Albert (1916), [http://www.alberteinstein.info/gallery/pdf/CP6Doc30_English_pp146-200.pdf "The Foundation of the General Theory of Relativity"] (PDF), Annalen der Physik 49

==References==
{{reflist}}

[[Category:Physics]]

Talk:General theory of relativity

2009-11-14T21:30:24Z

KSorenson: /* Idea for presenting this */

Talk:General theory of relativity

2009-11-14T21:29:29Z

KSorenson: /* Idea for presenting this */

General theory of relativity

2009-11-13T15:54:42Z

KSorenson: