There is a simple thought experiment in Newtonian gravity: drill a thin radial borehole all the way through a solid ball of uniform density, and drop a test particle into the hole, starting from rest at the very top. What happens?
The result is that the particle (blue, in the image on the left) falls all the way through the borehole, comes to a halt at the opposite end, then falls back, undergoing simple harmonic motion (the same kind of motion as an idealised version of a weight bouncing on the end of a spring) with exactly the same period as another test particle (red) orbiting the ball in a circular orbit that grazes the surface.
On this page, we will start by proving that result, but then we will also look at radial motion in the vacuum around the ball in Newtonian gravity, and examine how these systems work in General Relativity — featuring the famous Schwarzschild solution, but starring its much less famous cousin, the Other Schwarzschild solution!
A solid ball of radius r and uniform density ρ has mass:
m(r) = (4/3) π ρ r3
In Newtonian physics, the acceleration due to gravity from any spherically symmetrical arrangement of matter is directed towards the centre of symmetry, and it is the same as if all the matter that lies closer to the centre than the point where we are computing the acceleration was concentrated at the centre. (This result is known as the shell theorem, and it was proved by Newton himself.) So, anywhere inside our solid ball, the radial acceleration of a test particle is given by:
r''(t) = –G m(r(t)) / r(t)2
= –(4/3) G π ρ r(t)3 / r(t)2
= –(4/3) G π ρ r(t)
This is the same kind of equation as the one that governs a weight on a spring, where an object experiences a “restoring force” proportional to its displacement from some point. If the particle starts from rest at r = R at time t = 0, the solution is:
rsolid(t) = R cos(ω t)
where ω2 = (4/3) G π ρ
This is easily checked by taking the derivative of the claimed solution twice with respect to time.
How does this compare to the motion of a test particle orbiting a ball of radius R in a grazing circular orbit? If we equate the centrifugal acceleration for circular motion, ω2 R for a circle of radius R and an angular velocity of ω, with the inverse-square acceleration due to gravity, we have:
ω2 R = G m(R) / R2
ω2 = G m(R) / R3
= (4/3) G π ρ
So the angular velocity, and hence the period, is the same for the two kinds of motion.
We have shown that a test particle moving radially inside a ball of uniform density undergoes simple harmonic motion. But what about radial motion through the vacuum around such a ball?
If a test particle with unit mass starts from rest at some radius r1 which is greater than the radius of the ball, R, we can find its kinetic energy at any other r outside the ball from the change in its potential energy:
KE = ½ v(r)2
= –ΔPE
= G M (1/r – 1/r1)
where we are writing M for the total mass of the ball, m(R).
This is equivalent to:
dt/dr = 1/v(r) = ±1 / √[2 G M (1/r – 1/r1)]
The solution to this equation, for the particle starting its fall at time t = 0, is:
tvac(r) = √[r13/(2 G M)] (cos–1(√[r/r1]) + √[(r/r1)(1 – r/r1)])
We can compare this with the function giving time to reach a given radius for motion entirely through the solid ball:
tsolid(r) = √[R3/(G M)] cos–1(r/R)
In the limit where the ball is replaced by a point mass, the time for the test particle to fall down to r = 0 is:
tvac(0) = (π/2) √[r13/(2 G M)]
compared to:
tsolid(0) = (π/2) √[R3/(G M)]
This shows that the fall through the vacuum would be faster, which makes sense because the test particle feels the full force of all the mass throughout its motion, whereas inside the ball the amount of mass attracting it grows steadily smaller.
It is worth pointing out that the limiting case of a radial fall through the vacuum down to a point mass can be continued beyond r = 0 in two different ways. If we think of it as the limit of a series of ever skinnier elliptical orbits with the mass at one focus of the ellipse, in each case the test particle, at its closest approach, swings around the mass and reverses direction ever more rapidly. In that scenario, the fall down to r = 0 is half of a complete orbit, for an elliptical orbit whose semi-major axis a is half the starting radius, i.e. a = ½r1. The orbital period τ for a test mass in an elliptical orbit depends only on the orbit’s semi-major axis and the mass of the attracting body:
τ(a, M) = 2 π √[a3 / (G M)]
which give us, for half the orbital period of an orbit with a = ½r1:
½τ(½r1, M) = (π/2) √[r13/(2 G M)]
in agreement with the value we found for tvac(0).
This is in contrast with the scenario where we drop a test particle from the top of a radial borehole through a ball of uniform mass, but then fix the point where we drop the particle, while shrinking the ball and increasing its density to keep its mass constant. In that case, rather than swinging around the ball, the test particle always falls right through it. The time it takes to fall to r = 0 approaches the same limit, but when the acceleration becomes infinite the infalling solution continues as an ascending solution on the opposite side of the mass, rather than on the same side.
The plot above shows one cycle for various cases of the motion where a test particle falls from the same starting radius, r1, and passes through a borehole in a solid ball of the same mass, M, but different radii, R. The dots on each curve mark the extent of the solid ball. The slowest cycle takes place entirely within the ball, while the fastest is the limiting case where the ball has shrunk to a radius of zero but the test particle still “falls through it” rather than swinging around it.
In 1916, shortly before his death, the German physicist and astronomer Karl Schwarzschild published two papers that described exact solutions of Einstein’s equations for the geometry of space-time. The first solution is for the gravitational field due to a point mass[1], and this Schwarzschild solution is famous for describing the space-time geometry that we now understand to be that of a [non-rotating, electrically neutral] black hole. It also describes the vacuum surrounding any [non-rotating, electrically neutral] spherically symmetrical mass.
The second Schwarzschild solution is for the gravitational field due to a ball of incompressible fluid[2], or in other words a ball of uniform density, since an incompressible fluid does not change its density under pressure. So, this is the General Relativistic equivalent of the Newtonian solid ball of uniform density that we considered in the previous section.
The two Schwarzschild metrics, for a ball of total mass M and coordinate radius R, are given by[3]:
| ds2 | = | –(1–2M/r) dt2 + dr2 / (1–2M/r) + r2 (dθ2 + sin2θ dφ2) | r > R |
| ds2 | = | –¼(3 √[1–2M/R] – √[1–2Mr2/R3])2 dt2 + dr2 / (1–2Mr2/R3) + r2 (dθ2 + sin2θ dφ2) | r ≤ R |
Here we are following common practice in General Relativity and using geometric units, which are chosen to make the gravitational constant G and the speed of light c both equal to 1. In these units, times and masses are measured in the same units as distances. A time in conventional units is multiplied by c ≈ 2.998×108 metres/sec to convert it to a distance, and a mass is multiplied by G/c2 ≈ 7.425×10–28 metres/kg (or about 1,483 metres per solar mass).
The Schwarzschild r coordinate is chosen to make the surface area of a sphere of coordinate radius r equal to 4πr2, so the coordinates θ and φ for a given value of r act like ordinary spherical polar coordinates. The Schwarzschild t coordinate is chosen so that all slices of space-time with a fixed t coordinate have a geometry that is independent of t, as can be seen by the fact that t itself does not appear in the metric.
[1] “Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie” [On the gravitational field of a point mass according to Einstein’s theory] by K. Schwarzschild, Sitzber. Deut. Akad. Wiss. Berlin, Kl. Math.-Phys. Tech., pp 189-196, 1916.
[2] “Über das Gravitationsfeld einer Kugel aus inkompressibler Flussigkeit nach der Einsteinschen Theorie” [On the gravitational field of a sphere of incompressible fluid according to Einstein’s theory] by K. Schwarzschild, Sitzber. Deut. Akad. Wiss. Berlin, Kl. Math.-Phys. Tech., pp 424-434, 1916.
[3] Gravitation by Charles Misner, Kip Thorne and John Wheeler, W.H. Freeman, San Francisco, 1973. Section 23.7 and Box 23.2.
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