The first solution of the Einstein field equations is currently written as follows:
where
ds is an incremental length of an arbitrary line in space.
r[size=85]s[/size] is a constant. (It has a physical interpretation of the event horizon when dealing with black holes).
r is the length of a radial line from the center of a sphere (think of planets, stars, or black holes) to a small test object.
It should also be noted that all the variables including t and r are measured by an observer in an inertial reference frame theoretically at an infinite distance from the center of the sphere. Practically, the distance from the center of the sphere should be great enough that the gravitational pull is minimal compared with the effect being measured. (Since the Earth is not an inertial reference frame, this makes me suspicious about the confirmation results of the fabled advance of the perihelion of Mercury and the prediction of General Relativity. But that is another story).
This solution is attributed to Karl Schwartzchild, a brilliant physicist, who worked with David Hilbert at Gottingen during its’ glory days. This solution is for the particular case of a non rotating spherical body. Schwartzchild’s solution came in the same year as Hilbert first published the Einstein field equations.
If we constrain our view to the radial lines emanating from the center of a spherical body then:
Thus the equation for ds becomes:
Since light should travel along a geodesic, (Note that not all geodesics are the shortest distance between two points, but any curve representing the shortest distance between two points must be a geodesic) if we are to study the motion of light in a gravitational field, then ds = 0.
Thus we must have:
Here r[size=85]s[/size] is the Schwartzchild radius and for a non rotating spherical body and we can write:
Things to note:
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It is interesting that we can determine the speed of light, not by measuring how long it takes for light to go from point A to point B; but, instead, by looking at the required relationship of the variables on the surface of space.
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The field potential in a uniform field at a distance of r from the center of a mass would be given by:
This means Einstein’s uniform field solution is wrong by the factor of 2 in Equation 3. It should also be noted that in the natural world r is not fixed but varies from point to point. Therefore Equation 3 would only be practically valid over very small intervals.
- An object moving radially away from the center of spherical body will not be moving on a geodesic unless its’ speed is given by Equation 2.
If we look at the trivial case where M = 0, then we would be in Minkowsky space and the shortest path that light would take, would require that light move at the constant speed of c.
Since the location of the sphere was arbitrary and we are assuming that light moves on a geodesic path, we know that the speed of light is a constant c for all inertial reference frames. (This means that we don’t have to assume that the speed of light is constant in inertial reference frames, and we need not assume that the Lorentz transforms are valid. All we need to assume is that the Einstein field equations are valid, and that light moves along geodesic paths in order to show that light moves with a constant speed of c in inertial reference frames).
References:
Almost all of this information is available on Wiki, under the following:
Schwartzchild metric
Speed of light in a gravitational field
General Relativity
For the hardcore: “Spacetime and Geometry: An Introduction to Special Relativity” by Sean Carroll
There is, or was, an excellent treatment online by “Kevin Brown” but I can not find it right now.
The odd thing about Kevin Brown is that the name appears to be a pen name. No one seems to actually know who this person is. I like to think that he was the great right handed pitcher for the Los Angles Dodgers.
- Generally this action is not logically allowed. However, General Relativity assumes Infinitely differentiable functions, which effectively allows the differentials to be treated as simple numerical quantities.