6.7  Gravitational Acceleration in Schwarzschild Coordinates

 

If bodies, moved in any manner among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had not been urged by those forces.

                                                                                                Isaac Newton, 1687

 

According to Newton's theory the acceleration of gravity of a test particle at a given radial distance from a large mass is independent of the particle’s state of motion. Consequently it would be impossible to tell, from the relative motions of a group of free-falling test particles in a small region of space, that those particles were subject to any force. Maxwell emphasized the same point when he wrote (in the posthumously published “Matter and Motion”) that acceleration is relative, because only the differences between the accelerations of bodies can be detected.

 

Our whole progress up to this point may be described as a gradual development of the doctrine of relativity of all physical phenomena... There are no landmarks in space; one portion of space is exactly like every other portion, so that we cannot tell where we are. We are, as it were, on an unruffled sea, without stars, compass, soundings, wind, or tide, and we cannot tell in what direction we are going. We have no log which we can cast out to take a dead reckoning by; we may compute our rate of motion with respect to the neighbouring bodies, but we do not know how these bodies may be moving in space. We cannot even tell what force may be acting on us; we can only tell the difference between the force acting on one thing and that acting on another.

 

Of course, he was referring here to forces (such as gravity) that are proportional to inertial mass, so they impart equal accelerations to every body. As an example of a localized set of bodies subjected to equal acceleration, he considered ordinary objects on the earth’s surface, all of which are subjected (along with the earth itself) to the sun’s gravitational force and the corresponding acceleration. He noted that if this were not the case, i.e., if the sun’s gravity attracted only the earth but not ordinary small objects on the earth’s surface, this would be easily detectable by (for instance) changes in the position of a plumb line between sunrise and sunset.

 

Naturally these facts are closely related to the equivalence principle, but there are some subtle differences when we consider the accelerations of bodies due to gravity in the context of general relativity. We saw in Section 6.4 that the second derivative of r with respect to the proper time t of a radially moving particle in general relativity is simply

 

 

and thus independent of the particle’s state of motion, just as with Newtonian gravity. However, the proper times of two (momentarily) coincident particles may differ depending on their states of motion, so when we consider the motions of such particles in terms of a common system of coordinates the result will not be so simple. The second derivative of the radial coordinate r with respect to the time coordinate t in terms of the usual Schwarzschild coordinates depends not only on the spacetime location of the particle (i.e., r and t) but also on the trajectory of the particle through that point, even for particles with purely radial motion. To derive d2r/dt2 for purely radial motion we can divide through equation (1) of Section 6.4 by (dt)2 to give

 

 

Solving for dr/dt gives

 

 

where t is the proper time of the radially moving particle. We also have from Section 6.4 the relation

 

 

where k is a constant parameter of the given trajectory, and l is the path length parameter of the geodesic equations. We identify l with the proper time t by setting dt/dl = 1, so we can write

 

 

Substituting into (2), we have

 

 

and therefore the second derivative or r with respect to t is

 

 

In order to relate the parameter k to a particular trajectory, we can substitute (3) into equation (1), giving

 

 

There are two cases to consider. First, if there is a radius r = R at which the test particle is stationary, meaning dr/dt = 0, then

 

 

In this case the magnitude of k is always greater than 1. Inserting this into (4) gives

 

 

At the apogee of the trajectory, when r = R, this reduces to

 

 

as expected. If R is infinite, the coordinate acceleration reduces to

 

 

A plot of d2r/dt2 divided by -m/r2 for various values of R is shown below.

 

 

Notice that the value of (d2r/dt2) / (-m/r2) is negative in the range from r = 2m to r = 6m/(1 + 4m/R), where d2r/dt2 changes from negative to positive. This signifies that the acceleration (in terms of the Schwarzschild r and t coordinates) is actually outward in this range.

 

In the second case there is no radius at which the trajectory is stationary, so the trajectory escapes to infinity, and the speed dr/dt asymptotically approaches a fixed value V in the limit as r goes to infinity. In this case equation (5) gives

 

 

so the magnitude of k is less than 1. Inserting this into equation (4) gives

 

 

The case V = 0 corresponds to the case of R approaching infinity for the bound trajectories, and indeed we see that inserting V = 0 into this expression gives the same result as with R going to infinity in the acceleration equation for bound trajectories. At the other extreme, with V = 1, this equation reduces to

 

 

which is consistent with what we get for null (light-like) paths by setting dt = 0 in the radial metric and the solving for dr/dt = ±(1 – 2m/r). A normalized plot of this acceleration for various values of V is shown below.

 

 

We see that the acceleration d2r/dt2 in terms of the Schwarzschild coordinates r and t for a particle moving radially with ultimate speed V (either toward or away from the gravitating mass) is outward at all radii greater than 2m for all ultimate speeds greater than 0.577 times the speed of light. For light-like paths (V = 1), the magnitude of the acceleration approaches twice the magnitude of the Newtonian acceleration – and is outward instead of inward. The reason for this outward acceleration with respect to Schwarzschild coordinates is that the speed of light (in terms of these coordinates) is greater at greater radial distances from the mass.

 

Notice that the two expressions for d2r/dt2 derived above, applicable to the cases when the kinetic energy of the test particle is or is not sufficient to escape to infinity, are the same if we stipulate that R and V are related according to

 

 

If R is greater than 2m, then V2 is negative so V is imaginary. Hence in this case we find it most convenient to use R. On the other hand, if R is negative, from 0 to negative infinity, the value of V2 is real in the range from 0 to 1, so in this case it is convenient to work with V. The remaining possibility (which has no counterpart in Newtonian gravity) is if R is between 0 and 2m, in which case V2 is not only positive, it is greater than 1. Thus the impossibility of having a speed greater than 1 corresponds to the impossibility of being motionless at a radius less than 2m.

 

Incidentally, for a bound particle we can give an alternative derivation of the r,t acceleration from the well-known cycloidal parametric relations between r and t (derived in Section 6.4):

 

 

where R is the "top" of the orbit and q is an angular parameter that ranges from 0 at the top of the orbit (r = R) to p at the bottom (r = 0). Now, differentiating these parametric equations with respect to q gives

 

 

Therefore we have

 

 

If we now define the parameter u = cos(q) = 2r/R – 1, we have

 

 

Solving this for tan(q /2) gives

 

 

We want q = 0 at r = R so we choose the first root and substitute into the preceding equation for dr/dt to give

 

 

In addition, we have the derivative of coordinate time with respect to proper time of the particle

 

 

(See Section 6.4 for a derivation of this relation from the basic geodesic equations.) Dividing  dr/dt  by  dt/dt  gives

 

 

Just as we did previously, we can now compute d2r/dt2 = [d(dr/dt)/dr][dr/dt], and we arrive at the same result as before.

 

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