Linearized Einstein equations and a perturbation of the metric of the Minkowski space: superluminal communication
Albert Einstein in 1916 linearized his equations for a small perturbation h to the flat Minkowski space metric η:
If the stress-energy tensor T = 0, then we have:
where the box is the d'Alembert operator
□ = -1 / c² * d²/dt² + d²/dx² + d²/dy + d²/dz².
That is, each component of the perturbation h satisfies the familiar wave equation for light-speed waves. We use the East coast signature (- + + +) in the metric.
Now we see an immediate problem: if the metric of time is allowed to oscillate around the Minkowski metric, where
η₀₀ = -1,
then in some zones of spacetime, time flows faster than in the asymptotic Minkowski space. If the spatial metric is not stretched accordingly, then the speed of light defined in the global Minkowski coordinates would exceed c in those zones. That brings all the paradoxes of superluminal communication.
For electromagnetism, the analogous oscillation is not a problem. There are no paradoxes if the electric potential swings around the baseline potential of faraway space.
Could it be that the spatial metric is stretched enough to prevent superluminal messages?
The stretching should be in-sync with the undulation in the metric of time. If a gravitational wave is born from a binary star, we do not see why the spatial metric to every direction would be stretched in that way.
In the Schwarzschild metric, time has slowed down and the radial metric has been stretched. If the distorted metric sends a wave, we expect the metric of time to undulate, and there is also stretching of spatial metric to a certain direction at each point.
We conclude that linearized Einstein equations would produce metrics which are not satisfactory.
We do not know if Albert Einstein recognized this problem. Linearized equations allow the gravitational potential to swing above the potential of faraway space. That has similar consequences as matter of negative mass. Paradoxes abound.
How do you implement waves in a drum skin which is not allowed to swing above the horizontal level?
Should we change to comoving coordinates?
Wikipedia mentions the synchronous gauge which "requires that the metric does not distort measurements of time." Can that work?
Let us try to define comoving coordinates by putting a clock at each Minkowski spatial coordinate position
(n, m, l),
where n, m, l are integers.
y
^
| O O O clocks
|
| O O O
|
----------------------------------------> x
The clocks initially show the global Minkowski time. The positions of the clocks and the time which they show define comoving coordinates.
Let then a gravitational wave pass through the system of clocks.
What could go wrong? If the wave puts the clocks in a spatial disorder, then our spatial coordinates are of no use afterwards.
Also, if the wave leaves the clocks showing different times, then our coordinates are awkward. Traveling back in coordinate time would become possible.
Demetrios Christodoulou has shown that a wave can leave permanent changes in the relative positions of the clocks. It is called the gravitational wave memory effect.
Why would we define new coordinates? All physical phenomena stay exactly the same with the new coordinates. Also, if there are permanent changes in the relative positions or times shown by the clocks, the new coordinates become misleading.
We conclude that defining new coordinates is not a good idea.
The background metric around a wave packet
Could it be that nonlinearity somehow prevents superluminal communication inside gravitational waves?
The waves themselves carry mass-energy. If they bend the background metric in a suitable way, then the speed of light inside the waves might be slow enough to prevent superluminal communication.
However, if the wave propagates at the speed of light in the global Minkowski coordinates, it cannot slow down the speed of light toward the direction of its propagation.
This question is connected to the metric around a pulse of light. People believe that the background metric around a packet of gravitational waves is similar to the metric around a wave packet of light. Let us write another blog post about this question.
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