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/* The right side of the equation: the stress-energy tensor */
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<math>T_{\mu\nu}</math>=== 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.
===The left side of the equation: the Einstein curvature tensor===
