General theory of relativity

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This article/section deals with mathematical concepts appropriate for a student in late university or graduate level.

The general theory of relativity is a theory of gravity that is compatible with 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, 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."

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. 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

This article/section deals with mathematical concepts appropriate for late high school or early college.

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 geodesics 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.

In this form, represents the Einstein tensor, is the same gravitational constant that appears in the law of universal gravitation, and is the stress-energy tensor (sometimes referred to as the energy-momentum tensor). The indices and range from zero to three, representing the time coordinate and the three space coordinates in a manner consistent with 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 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, , 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 component of 4-momentum across a surface of constant coordinate .

Fine. But what does that mean?

In classical mechanics, it's customary to refer to coordinates in space as , and . In general relativity, the convention is to talk instead about coordinates , , , and , where is the time coordinate otherwise called , and the other three are just the , and coordinates. So "a surface of constant coordinate "" simply means a 3-plane perpendicular to the 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 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.

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

energy density, which is equivalent to mass-energy density; this component includes the mass contribution

, ,

the components of momentum density

, ,

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:

, , , , ,

the components of shear stress, or stress applied tangential to the region

, ,

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 , , and , 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 — , and — 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 . 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.

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 next section.

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 . 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.

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), "The Foundation of the General Theory of Relativity" (PDF), Annalen der Physik 49