Let us assume that the metric around M really describes the force on m.
The change of the metric required to reduce the attraction of M
m • ● M
^ y
|
------> x
The test mass m is initially static. What kind of a change in the metric could decrease the acceleration
d²x / dτ²,
where x is the x coordinate of m, and τ is the proper time of m?
In the geodesic formula, μ is the x coordinate and α and β must be the t coordinate. All the other terms are zero.
We have
Γₓₜₜ = 1/2 (2 dgₓₜ / dt - dgₜₜ / dx).
We can decrease the force on m by making gₜₜ less steep in the x direction, or making gₓₜ to increase with time.
What would that require to happen in the stress-energy tensor T of the vacuum area? The tensor T cannot remain zero because then Schwarzschild metric around M could not change at all.
The element gₜₜ becomes less steep if M ejects mass-energy out and the ejected mass-energy passes m.
We can manipulate gₓₜ by letting a compact mass M' pass by m at a close distance. But how could a spherically symmetric process do that?
Let us calculate the Einstein tensor for the metric g. From that we get the required stress-energy tensor T. Or do we? The component gₜₜ can be made steeper by moving M closer to m. Many different changing stress-energy tensors may be able to bring the desired changes.
This looks complicated.
Ejection of negative mass-energy?
Can we make the metric to change with time? Yes, if we assume that some strange matter whose mass-energy is negative is ejected from M when we increase the pressure.
A shell of negative mass-energy would recede from M, making the effective mass of M larger. But negative mass-energy may facilitate superluminal communication and bring all the time paradoxes. This is not a nice solution.
Also, if we want to reduce the attraction of M, we should eject a shell of positive mass-energy. What kind of matter would that be?
Can small departures from perfect symmetry help?
The system is never perfectly symmetric. Can we utilize small departures from the symmetry to manipulate the metric in such a way that it exerts a varying force on m, depending on the time?
How "stable" is Birkhoff's theorem? Could it be that the metric could otherwise change with time, but the perfect symmetry prevents changes in the metric from "escaping" from M to empty space?
This idea is against our own Minkowski-newtonian gravity model. The gravity field in it is determined in a simple fashion from the sources of gravity. Small departures from symmetry cannot make much of a difference.
Since the Einstein field equations are nonlinear, it is very difficult to prove mathematically that small perturbations cannot work miracles. But we may demand that a physical theory must not be based on almost-miracles.
The Schwarzschild interior solution when the pressure is suddenly released
Karl Schwarzschild in 1916 was able to glue together an interior solution for incompressible fluid ball and the external solution in the vacuum.
What would happen if we would suddenly remove the pressure which is stopping the incompressible fluid from collapsing? The stress-energy tensor T inside the ball suddenly changes. How does that affect the vacuum solution?
Birkhoff's theorem claims that the vacuum solution cannot change. The metric should solely start changing inside the ball. Is that possible?
In the Einstein approximation formula (1916), pressure does contribute to the metric in the vacuum area. The pressure P is "mapped" to the point we are interested in through the formula
~ P / r,
where r is the distance from the point to the pressurized element. After trace reversing, the pressure affects gₜₜ, gₓₓ, and so on, at the point.
A pressure which changes with time
The stress-energy tensor of a particle moving in the x direction looks like this:
Dirac δ function *
γ m γ v m 0 0
γ v m γ v² m 0 0
0 ...
...
We can "simulate" pressure with moving matter.
• --> <-- •
<-- • ... • -->
Pressure T₁₁, T₂₂, or T₃₃ in the stress-energy tensor can come either from matter moving to opposite directions, or from mechanical pressure from a force field.
However, we cannot manipulate T₀₀, but only the pressure and the mass-energy flow components. It could still be that the pressure does not cause any effect on a test mass m which is outside a ball of matter M.
The Ehlers et al. paper (2005) proves that one can preserve the metric outside the ball, if there is a negative pressure in a "crust" of the ball. What happens if we remove the crust?
Let us think. The Schwarzschild interior solution proves that it is possible to find a metric for a pressure which is quite a complicated function of the radius r. The external metric is the Schwarzschild vacuum solution.
Is there any reason why a time-dependent pressure should affect the metric in the vacuum area?
Here we must note the following three things:
1. Pressure probably does affect the orbit of a pointlike test mass m outside the spherical mass M. We have in this blog repeatedly argued that "tidal" effects in M and a "backreaction" change the orbit of m.
2. But we have also repeatedly argued that the metric derived from the Einstein field equations does not describe the orbit of the test mass m. That is, the geodesic equation fails when there are tidal effects.
3. It is possible that a pressure inside M does not affect the vacuum metric calculated from the Einstein field equations, but the pressure does affect the orbit of m.
The Einstein equations may be flexible enough so that they allow the metric inside M to change when the pressure is changed, but do not require the metric outside M to change.
The pressure is always zero at the surface of M. It is not far-fetched to assume that there is no need to change the metric outside M when we change the metric inside M, to suit a pressure change.
In an earlier blog post we suggested that in a spherically symmetric configuration, the test mass m actually has to be a spherical shell. The test mass must not break the spherical symmetry. A spherical shell, apparently, does not distort the spatial metric inside it – only the metric of time. Pressure inside a spherical mass M would not attract a spherical shell outside M.
A pressure "focuses" beams of light passing through M: it must bend light also outside M?
In general relativity, mass-energy, and presumably also pressure, "focuses" a narrow beam of light if there is mass or pressure inside the volume of the beam. Empty space does not focus a beam of light.
--------------------------------
----------_______________ beam of light
/ \
\____/ M
Suppose that M only bends rays of light which touch M or pass through M. Then there is "defocusing" of a beam which touches M. This is not allowed.
In the Schwarzschild solution, the focusing capability is only inside M, but the metric also bends rays of lights which do not pass through M.
Now, if we are able to increase the focusing capability of M temporarily with an increased pressure, we expect that M should also bend rays of light outside M more strongly. But Birkhoff's theorem prohibits that!
In the Einstein approximation formula (1916), pressure does affect the metric also outside M.
If we are able to prove that an increased pressure inside M break's Birkhoff's theorem, that might imply that the Einstein field equations do not have a solution for a realistic physical system.
Conclusions
Our analysis converged on a crucial question: does a changed pressure inside a spherically symmetric mass M require that the metric outside M changes. That would break Birkhoff's theorem.
We have to check what happens if we remove the pressure from the Schwarzschild internal solution. Also, we have to study how focusing/defocusing are precisely defined for a beam of light.
If Birkhoff's theorem is broken, that may imply that the Einstein field equations do not have a solution for any realistic physical system.
No comments:
Post a Comment