G4 Clocks and Yardsticks in the Schwarzschild Field
Gravitational fields of non rotating spherical masses are often called Schwarzschild fields in honor of the German physicist and astronomer Karl Schwarzschild (1873-1916). Schwarzschild was concerned at the turn of the century with the question of whether the physical space of astronomy is really Euclidean or not. He had already begun in 1913 to look for the redshift of spectral lines of the sun predicted by Einstein. A few weeks after the publication of Einstein's equations, he was the first to present an exact solution, and some weeks later, he delivered a second solution.
He wrote these works while serving in the war on the Eastern Front where he contracted a skin disease from which he died in 1916.
Karl Schwarzschild (1873-1916)
Let us once again consider our little laboratory, as in the last section, starting from the 'OFF' position and falling toward a spherical mass. We want to compare the measurements of an observer in this laboratory to those of a non-moving observer in OFF position, that is, to an observer at rest and at a very large distance from M (and any other large mass).
According to the equivalence principle the freely falling laboratory is at every moment a gravitational-free inertial frame in the sense of the STR. Thus we already know the relationship between the measurements made in the falling laboratory which is at a distance r from the center of M to those made by the observer in ‘OFF’ position: r together with α (or Rs or Φ) determine the value of our radical:
∆t(r) = ∆t(∞) • √ = ∆t(∞) • √( 1 - Rs / r ) ≈ ∆t(∞) • ( 1 - α / r ) = ∆t(∞) • ( 1 + Φ(r) / c2 )
∆x(r) = ∆t(∞) / √ = ∆x(∞) / √( 1 - Rs / r ) ≈ ∆x(∞) / ( 1 - α / r ) = ∆x(∞) / ( 1 + Φ(r) / c2 )
∆y(r) = ∆y(∞) (no lateral contraction) Rs
∆z(r) = ∆z(∞) (no lateral contraction)
We obtain these results from the STR and our fourth formulation of the equivalence principle. With free fall we have caused gravitation to completely disappear, replacing it for the observer in the OFF position with a continually adjusted acceleration. The occupants in the free falling laboratory are in fact in an inertial frame the whole time. They must, for example, always measure the same wavelengths in the spectrum of an excited hydrogen atom (the experiment taking place entirely within the falling laboratory).
Assume now they fly past the place r with the instantaneous velocity v = √ (2 • G • M / r). In so doing they measure the frequencies in a glowing hydrogen gas cloud which is there at rest in distance r of M. After taking into account the Doppler effect (the laboratory occupants know their STR) they must obtain the usual well-known values because being in their laboratory system they are indeed in an inertial frame. No such thing as gravitation exists for them!
So the hydrogen gas resting at site r in the gravitational field radiates at frequencies which are measured correctly with the clocks of our free falling laboratory. But these clocks are slow running as seen from the OFF position! This means, however, that clocks (or any other oscillating systems) at rest at a fixed distance r from the center of gravity run slower (in comparison with those in the OFF) by exactly the same factor as those in our falling laboratory.
We thus arrive to the following formulation of the equivalence principle, which no longer applies generally, but is instead tailored to our specific situation:
Measurements of lengths and time intervals, in a small laboratory at rest at place r in the gravitational field of M are transformed the same as those of our freely falling laboratory falling past point r along a field line.
Considering the above equations and those from the last section, we can derive the following qualitative statements:
- The smaller r is, the less time elapses when compared to a clock in OFF. The stronger the gravitational field is, the slower the clocks run! Clocks at the same distance from the center of M run equally fast. Rs
- The smaller r is, the longer a segment in the radial direction will be when measured with local yardsticks. As seen from OFF: yardsticks shorten in the radial direction with increasing strength of the gravitational field! Thus, for the thickness of a spherical shell around M, a local surveyor determines a larger value than an observer in OFF. Rs
- The circumference of a circle around the center of M will be measured equally by local observers and by an observer from OFF.
The second and third points together imply that for an observer in the gravitational field the diameter of a circle is longer than its circumference divided by π ! Therefore the laws of Euclidean geometry no longer apply in a gravitational field. The fact that the diameter is longer than expected (by Euclidean geometry) when measured locally but not when measured from OFF, is usually illustrated as follows:
At the top: A dimple or depression is drawn that has the property that the diameter measured along the gray area is longer than the circumference divided by π . One happily lets a planet circle around this dimple, as if it had a top and bottom and an additional gravitational field in the z-direction!
Epstein complains vigorously about this “powerfully misleading notion” [15-169]. He is right. With this dimple one is attempting to show only the metric in the equatorial plane of the star. The yardsticks always lie in the equatorial plane (middle image), but they shorten, when one approaches the surface of the star (and in the center of the star have the same length as in OFF).
To depict this distortion with respect to the Euclidean metric, one extends the equatorial plane into ‘hyperspace’ - whether you have a dimple pointing ‘down’ or a bump pointing ‘up’ is irrelevant (bottom picture). This additional dimension has nothing to do with the z-direction. The point of inflection of the cross-section of this dimple has the z-axis distance R, where R is the radius of the star as measured from OFF.
For the following, assume we are sitting next to an observer in OFF, that is, very far from the mass M, at a place where the potential Φ(r) practically disappears (that is, it approaches zero...). This slightly fictitious position helps when we consider transforming the readings from a laboratory at distance r1 into those of a laboratory at a distance r2 from the center of mass. Imagine, for example, a flashing light of constant frequency at point r1. What time interval does a local observer at point r2 measure with his clock until he has counted 100 light flashes?
According to the formulas at the beginning of this section we have
∆t(r1) / √( 1 - Rs / r1 ) = ∆t(∞) = ∆t(r2) / √( 1 - Rs / r2 ) mmm and thus
∆t(r2) = ∆t(r1) • √( 1 - Rs / r2 ) / √( 1 - Rs / r1 ) = ∆t(r1) • √( ( 1 - Rs / r2 ) / ( 1 - Rs / r1 ) )
Similarly, for small lengths in the radial direction (i.e., in the x-direction)
∆x(r2) • √( ( 1 - Rs / r2 ) = ∆x(∞) = ∆x(r1) • √( 1 - Rs / r1 ) mm etc.
(Small) segments orthogonal to the field lines are, however, for all observers of equal length.
Using our approximations we get the following simpler results:
∆t(r2) / ( 1 - α / r2 ) ≈ ∆t(∞) ≈ ∆t(r1) / ( 1 - α / r1 )
∆x(r2) • ( 1 - α / r2 ) ≈ ∆x(∞) ≈ ∆x(r1) • ( 1 - α / r1 )
∆t(r2) / ( 1 + Φ(r2) / c2 ) ≈ ∆t(∞) ≈ ∆t(r1) / ( 1 + Φ(r1) / c2 )
∆x(r2) • ( 1 + Φ(r2) / c2 ) ≈ ∆x(∞) ≈ ∆x(r1) • ( 1 + Φ(r1) / c2 )
Thus, we know exactly how the location-dependent readings for time intervals and lengths must be transformed. Let’s treat ourselves to a small calculation: A clock on top of a 22.6 meter high tower emits exactly one beep every second. An identical clock sets at the foot of the tower and measures the time between the beeps. We already know that the lower clock runs slightly slower. What time interval between the beeps does the lower clock measure?
For the ratio Δt(top) / Δt(bottom) we obtain, using the above formulas, the expression
√( ( 1 - 2•αE / (rE + 22.6) ) / ( 1 - 2•αE / rE ) ) mm with αE ≈ 4.43•10-3 m and rE ≈ 6.373•106 m
Entering these values in most calculators will yield simply 1! The two values differ so little that one cannot, even with 10 or 12 digits, see a difference with 1! The differences in small shifts of the gravitational field of the earth are so small that it borders on a miracle that they were already measured in 1960 (experiment of Pound and Rebka, see I4). In order to calculate the size of the effect, we should therefore not form the ratio Δt(top) / Δt(bottom), but rather form the ratio of the tiny difference between the two periods to one of the times itself. We would then get something that is smaller than 10-12.
Thus we determine ( ∆t(top) - ∆t(bottom) ) / ∆t(bottom) = ( ∆t(r2) - ∆t(r1) ) / ∆t(r1)
It is recommended to use the approximation formula without the radical:
rE2 for r2 • (r1 - α). Thus, we get the very simple result
We come to a more familiar realm when we do the same calculation with the potential. As before we quickly get (starting with the approximation without the radical) the first term in the following line:
The first simplification is based on our additional restriction, which is easily met in the gravitational field of the earth: Φ(r) is everywhere much smaller than c2 and can be omitted in the denominator. And for the small height difference of 22.6 m at the earth's surface we can use the good approximation g • Δh for the potential difference in the numerator! With the input 9.81 • 22.6 / 9 • 1016 even our calculator is well-behaved giving the result 2.46 • 10-15.
We have just quantitatively mastered a famous test of GTR ! The real trick is, firstly, to formulate the problem in a manner that allows a convenient calculation and, secondly, to cleverly use approximations so the result can actually be determined. More examples will follow.