Inverting The Schwarzschild Transformation

In the Schwarzschild Transformation we have a pair of equations (actually four, two of which are suppressed because they are irrelevant) that translate measurements of two events made by an observer close to a simple gravitating body (using lower-case variables) into the same measurements that would be made of the same events by an observer far from the gravitating body (using upper-case variables). Because the large simple body exerts a gravitational force, there exists a nonzero virtual relative velocity between the observers, even though there exists (by assumption) no actual velocity between them. As a consequence of that fact, the translation of measurements must include relativistic effects, in particular the appropriate analogues of time dilation and the Lorentz-Fitzgerald contraction.

As in Special Relativity, the virtual relative velocity between the observers must have the same value (though opposite algebraic sign) for both observers. That fact necessitates that the Lorentz factor between the observers have the same value for both of them; that is,

(Eq地 1)

In this case the squared velocity comes from imaging a small body falling from the upper observer to the lower observer and equating the kinetic energy that it gains to the gravitational potential energy that it loses, which leads us to

(Eq地 2)

in which m represents the mass of the large body and r represents the distance between the lower observer and the center of the large body, both measured by the lower observer. Combining that equation with Equation 1 leads to

(Eq地 3)

Because the lower observer comes arbitrarily close to the large body, there is effectively no virtual relative velocity between that observer and the body, so m represents, to arbitrary precision, the rest mass of the large body. A virtual relative motion does exist between the upper observer and the large body, so the large body must ponder more mass for the upper observer in accordance with the relativistic mass increase,

(Eq地 4)

In that equation the equivalent of the Lorentz factor comes from combining Equations 1 and 2 and Equation 3 tells us that it has the same value, whether we use m/r or M/R. Comparing that result with Equation 3 tells us that

(Eq地 5)

which necessitates that

(Eq地 6)

That痴 the first equation of the semi-classical Schwarzschild Transformation.

The wave-particle duality of the quantum theory tells us that we must associate the mass-energy of a particle with a wave of a frequency proportional to that mass-energy in accordance with Planck痴 theorem,

(Eq地 7)

in which dt represents the period of the wave that would be measured by the observer. If that equation represents measurements made by the lower observer, then a comparison with Equation 4 gives us the second equation of the Schwarzschild Transformation directly,

(Eq地 8)

I have not included the longitudinal and latitudinal dimensions in this analysis. They don稚 differ between our two observers, so there痴 no reason to include them. They would actually be a distraction and are better left out here.

We can represent the Schwarzschild Transformation as a matrix multiplication with the transformation matrix derived directly from Equations 6 and 8:

(Eq地 9)

In that equation I multiplied the temporal coordinates by the speed of light so that all of the measurements would have units of length. If we multiply that equation by the inverse of the transformation matrix, we will get back the measurements that we started with;

(Eq地 10)

Thus we must have the product of the two matrices as

(Eq地 11)

the identity matrix. We can see by simple inspection that

(Eq地 12)

so we have the inverted transformation equations as

(Eq地s 13)

In those equations I have again made use of the fact that m/r=M/R.

If we multiply the differential spatio-temporal four-vector by itself, we get the metric equation, the four-dimensional analogue of the Pythagorean theorem:

(Eq地 14)

The matrix on the third line of that equation is the metric tensor, which I
obtained by multiplying the second matrix on the second line on the left by the
transpose of the first matrix. In that equation, on the second line, I included
the imaginary coefficient (the square root of minus one) on the temporal
components so that a minus sign would automatically appear in the right place in
the metric equation and then I shifted them into the g_{44} element of
the metric tensor as a factor of minus one. We also have the inverted version of
the metric equation from Equations 12 and 13:

(Eq地 15)

But that痴 all semiclassical. It only applies in regions of space whence escape velocity from the large body is very much less than the speed of light. If we want to consider regions deep inside a stiffer gravitational field, regions whence escape velocity from the large body approaches a large fraction of the speed of light, then we must use the fully relativistic version of the Schwarzschild solution of Einstein痴 equation.

The relativistic equivalent of Equation 2 appears as

(Eq地 16)

In that equation I merely divided out the rest mass of the dropped object. Thus we get the Lorentz factor directly as

(Eq地 17)

That result lets us write the fully relativistic equivalents of Equations 6 and 8 as

(Eq地s 18)

The metric equation that corresponds to that transformation looks like this:

(Eq地 19)

We have the inverses of Equations 18 as

(Eq地s 20)

The corresponding metric equation comes to us as

(Eq地 21)

Thus we gain a slightly deeper understanding of the Schwarzschild solution of Einstein痴 equation and of the spacetime that it describes.

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