Physics in a curved spacetime

manifold, but the Euclidean space had its usual metric and time was measured by a universal clock. Observers with different velocities were all equally valid: this form of relativity was built into Galilean mechanics. Therefore there was no universal standard of rest, and different observers would have different definitions of whether two events occurring at different times happened at the same location. But all observers would agree on simultaneity, on whether two events happened in the same time-slice or not. Thus the ‘separation in time’ between two events meant the time elapsed between the two Euclidean slices containing the two events. This was independent of the spatial locations of the events, so in Newtonian gravity there was a universal notion of time: all observers, regardless of position or motion, would agree on the elapsed time between two given events. Similarly, the ‘separation in space’ between two events meant the Euclidean distance between them. If the events were simultaneous, occurring in the same Euclidean time-slice, then this was simple to compute using the metric of that slice, and all observers would agree on it. If the events happened at different times, each observer would take the location of the events in their respective space slices and compute the Euclidean distance between them. The locations would differ for different observers, but again the distance between them would be the same for all observers.

However, in Newtonian theory there was no way to combine the time and distance measures: there was no invariant measure of the length of a general curve that changed position and time as it went along. Without an invariant way of converting times to distances, this was not possible. What Einstein brought to relativity was the invariance of the speed of light, which then permits a unification of time and space measures. Einstein’s four-dimensional spacetime has a much simpler structure than Newton’s!

Now, within this model of spacetime, Newton gave a law for the behavior of objects that experienced gravitational forces: F = ma, where F = −m φ for a given gravitational field φ. And he also gave a law determining how φ is generated: 2 = 4π Gρ. These two laws are the ones we must now find analogs for in our relativistic point of view on spacetime. The second one will be dealt with in the next chapter. In this chapter, we ask only how a given metric affects bodies in spacetime.

We have already discussed this for the simple case of particle motion. Since we know that the ‘acceleration’ of a particle in a gravitational field is independent of its mass, we can go to a freely falling frame in which nearby particles have no acceleration. This is what we have identified as a locally inertial frame. Since freely falling particles have no acceleration in that frame, they follow straight lines, at least locally. But straight lines in a local inertial frame are, of course, the definition of geodesics in the full curved manifold. So we have our first postulate on the way particles are affected by the metric:

(IV) Weak Equivalence Principle: Freely falling particles move on timelike geodesics

of the spacetime.1

1It is more common to define the WEP without reference to a curved spacetime, but just to say that all parti-

cles fall at the same rate in a gravitational field, independent of their mass and composition. But the Einstein Equivalence Principle (Postulate IV ) is normally taken to imply that gravity can be represented by spacetime

curvature, so we shall simply start with the assumption that we have a curved spacetime.

By ‘freely falling’ we mean particles unaffected by other forces, such as electric fields, etc. All other known forces in physics are distinguished from gravity by the fact that there are particles unaffected by them. So the Weak Equivalence Principle (Postulate IV) is a powerful statement, capable of experimental test. And it has been tested, and continues to be tested, to high accuracy. Experiments typically compare the rate of fall of objects that are composed of different materials; current experimental limits bound the fractional differences in acceleration to a few parts in 1013 (Will 2006). The WEP is therefore one of the most precisely tested laws in all of physics. There are even proposals to test it up to the level of 10−18 using satellite-borne experiments.

But the WEP refers only to particles. How are, say, fluids affected by a nonflat metric? We need a generalization of (IV):

(IV ) Einstein Equivalence Principle: Any local physical experiment not involving gravity will have the same result if performed in a freely falling inertial frame as if it were performed in the flat spacetime of special relativity.

In this case ‘local’ means that the experiment does not involve fields, such as electric fields, that may extend over large regions and therefore extend outside the domain of validity of the local inertial frame. All of local physics is the same in a freely falling inertial frame as it is in special relativity. Gravity introduces nothing new locally. All the effects of gravity are felt over extended regions of spacetime. This, too, has been tested rigorously (Will 2006).

This may seem strange to someone used to blaming gravity for making it hard to climb stairs or mountains, or even to get out of bed! But these local effects of gravity are, in Einstein’s point of view, really the effects of our being pushed around by the Earth and objects on it. Our ‘weight’ is caused by the solid Earth exerting forces on us that prevent us from falling freely on a geodesic (weightlessly, through the floor). This is a very reasonable point of view. Consider astronauts orbiting the Earth. At an altitude of some 300 km, they are hardly any further from the center of the Earth than we are, so the strength of the Newtonian gravitational force on them is almost the same as on us. But they are weightless, as long as their orbit prevents them encountering the solid Earth. Once we acknowledge that spacetime has natural curves, the geodesics, and that when we fall on them we are in free fall and feel no gravity, then we can dispose of the Newtonian concept of a gravitational force altogether. We are only following the natural spacetime curve.

The true measure of gravity on the Earth are its tides. These are nonlocal effects, because they arise from the difference of the Moon’s Newtonian gravitational acceleration across the Earth, or in other words from the geodesic deviation near the Earth. If the Earth were permanently cloudy, an Earthling would not know about the Moon from its overall gravitational acceleration, since the Earth falls freely: we don’t feel the Moon locally. But Earthlings could in principle discover the Moon even without seeing it, by observing and understanding the tides. Tidal forces are the only measurable aspect of gravity.

Mathematically, what the Einstein Equivalence Principle means is, roughly speaking, that if we have a local law of physics that is expressed in tensor notation in SR, then its mathematical form should be the same in a locally inertial frame of a curved spacetime.

This principle is often called the ‘comma-goes-to-semicolon rule’, because if a law contains derivatives in its special-relativistic form (‘commas’), then it has these same derivatives in the local inertial frame. To convert the law into an expression valid in any coordinate frame, we simply make the derivatives covariant (‘semicolons’). It is an extremely simple way to generalize the physical laws. In particular, it forbids ‘curvature coupling’: it is conceivable that the correct form of, say, thermodynamics in a curved spacetime would involve the Riemann tensor somehow, which would vanish in SR. Postulate (IV ) would not allow any Riemann-tensor terms in the equations.

As an example of how (IV ) translates into mathematics, we discuss fluid dynamics, which will be our main interest in this course. The law of conservation of particles in SR is expressed as

where n is the density of particles in the momentarily comoving reference frame (MCRF), and where Uα is the four-velocity of a fluid element. In a curved spacetime, at any event, we can find a locally inertial frame comoving momentarily with the fluid element at that

event, and define n in exactly the same way. Similarly we can define U to be the time basis vector of that frame, just as in SR. Then, according to the Einstein equivalence principle (see Ch. 5), the law of conservation of particles in the locally inertial frame is exactly Eq. (7.1). But because the Christoffel symbols are zero at the given event because it is the origin of the locally inertial frame, this is equivalent to

This form of the law is valid in all frames and so allows us to compute the conservation law in any frame and be sure that it is the one implied by the Einstein equivalence principle. We have therefore generalized the law of particle conservation to a curved spacetime. We will follow this method for other laws of physics as we need them.

Is this just a game with tensors, or is there physical content in what we have done? Is it possible that in a curved spacetime the conservation law would actually be something other than Eq. (7.2)? The answer is yes: consider postulating the equation

where R is the Ricci scalar, defined in Eq. (6.92) as the double trace of the Riemann tensor, and where q is a constant. This would also reduce to Eq. (7.1) in SR, since in a flat spacetime the Riemann tensor vanishes. But in curved spacetime, this equation predicts something very different: curvature would either create or destroy particles, according to the sign of the constant q. Thus, both of the previous equations are consistent with the laws of physics in SR. The Einstein equivalence principle asserts that we should generalize Eq. (7.1) in the simplest possible manner, that is to Eq. (7.2). It is of course a matter for experiment, or astronomical observation, to decide whether Eq. (7.2) or Eq. (7.3) is correct. In this book we shall simply make the assumption that is nearly universally made, that the Einstein equivalence principle is correct. There is no observational evidence to the contrary.