UPDATE November 20, 2021: We added a link to the Physics stack exchange question where an author claims that the metric of time cannot change in a gravitational wave. We added a link to a paper by A. Loeb and D. Maoz where practical measurements of the oscillating metric of time are discussed.
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The limiting case of a long rod of ordinary matter moving at almost the speed of light
long rod
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----> v ≅ c
Our first guess, naturally, is that the metric is similar to ever longer rods of ordinary matter which we make to move ever faster.
The length of the rod is kept constant, say, 1 meter in the frame of the observer. The total energy of the rod is kept constant, say,
E = m c².
The electric field of a fast-moving charge is squeezed in the direction of its movement. The newtonian gravity field of the rod might be cylindrical?
Let us put an initially static test mass close to the orbit of the rod. The rod feels the gravity field of the test mass long before the test mass knows anything of the approaching rod and the orbit of the rod starts to bend.
Momentum has to be conserved. When the rod has passed, the test mass must have received the momentum which the rod exchanged with the gravity field of the test mass.
A cylindrical field might be the right solution.
Our own syrup model of gravity suggests that the rod makes test pulses of light to travel along it. The speed of a test pulse in the global Minkowski coordinates is almost c to the direction of the rod movement. The speed of light may be considerably less than c to any other direction.
What kind of a metric could describe this behavior?
The rod pulls on the test mass. The flow of time must be slower than the Minkowski time close to the rod, to implement the pulling force. For slow speeds of the rod, this is certainly true.
Comoving clocks versus static clocks close to the rod
Let us have an observer comoving with the rod at almost the speed of light. He sees the rod as very long, and its mass is very small. He thinks that the metric is very close to the flat Minkowski metric. He sees that the speed of light is slightly below c close to the rod, to every direction. The radial metric is slightly stretched.
The comoving observer sees the mass of the rod as
m / γ
where
γ = 1 / sqrt(1 - v² / c²).
If the comoving observer shoots a ray of light to the left in the diagram, past the rod, the ray is only deflected by an angle
α ~ m / γ
by the gravity of the rod. However, because of length contraction, a static observer sees the angle as
α' ~ m.
This makes sense: the static observer sees the whole mass-energy of the fast moving rod to deflect the light.
What about the flow of time? If there is a clock attached to the rod, the comoving observer sees it tick only slightly slower than his own clock, because the mass is only m / γ.
If we have a static observer normal to the rod very far away observing time signals from both clocks, he sees both clocks ticking very slowly, and the clock attached to the rod ticking just slightly slower than the comoving clock.
What about a static observer close to the rod? How much has his time slowed down?
If a static clock very close to the rod send a signal, there is a considerable redshift relative to a static clock far away from the rod.
The relative redshift in the comoving clocks is much less than in the static clocks. How can we explain this?
It is probably frame dragging. The "effective" velocity of the clock attached to the rod is less than one would expect because the rod drags the frame along with it. If we have a rotating large mass, we can make a clock close to the mass to tick faster by letting it comove with the mass.
The mass-energy of a gravitational wave packet
Time for a static observer is slowed down considerably near the rod. We want to find out if this slowdown is enough to cancel the speeding up of time in the "crests" of the wave. That is, if clocks close to the wave never tick faster than clocks far away in the Minkowski space.
S. V. Babak and L. P. Grishchuk calculate in the link the stress-energy pseudotensor for a perturbation h of the Minkowski space metric. It is the formula (27) in the paper, and we see that every term is proportional to the product of two partial derivatives of components of h.
The speedup of time is proportional to h₀₀. If we divide h by a large number N, then the energy density is only 1 / N². We immediately see that the potential generated by the energy density cannot cancel a possible speedup of time in a gravitational wave.
What about frame dragging? Could it cause the signal to take a longer path, so that communication cannot be too fast? If frame dragging is proportional to the mass-energy of the wave, then it cannot slow it enough. Also, frame dragging could be used to move the signal to the right direction. Then it would not slow down communication.
What about a gravitational wave which contracts spatial metric in the Minkowski space?
Let us image that we have an orthogonal spatial coordinate grid drawn into the Minkowski space.
Let a gravitational wave packet pass by and contract the spatial distance between two observers A and B. If A sends a light signal to B during that time, then the signal appears to have moved faster than light. Does this bring us all the paradoxes of superluminal communication?
We may imagine that the wave just temporarily moves A and B, and the true metric remains exactly the flat Minkowski metric. Then there is no paradox.
However, if A and B are not inside the wave packet, and the distance anyway gets contracted, then we have true superluminal communication which brings the paradoxes.
https://physics.stackexchange.com/questions/553487/do-gravitational-waves-cause-time-dilation-or-not
In the answers to the Physics stack exchange question (2020) above, an author Paul T. claims that a gravitational wave changes the spatial metric but not the metric of time. That is a strange claim. If we in the Minkowski space in the frame 1 have a spatial distance, then in a moving frame 2 the distance is both spatial and temporal.
Paul T. writes that by "gauge fixing" we can show that the metric of time is constant. But the choice of the gauge cannot affect observed physical phenomena. If someone observes that clocks tick at different rates, that cannot be altered in any gauge.
Several authors have a consensus that the component h₀₀ and other components of the perturbation h obey the standard wave equation in linearized Einstein equations. Thus, there are waves in the metric of time.
Abraham Loeb and Dan Maoz (2015) write about observing mHz gravitational waves by distributing atomic clocks to the Solar system.
The consensus seems to be that gravitational waves do affect the metric of time.
Conclusions
If a gravitational wave allows superluminal communication in an asymptotic Minkowski space, we get all the causality paradoxes. Breaking causality is not accepted in a robust physical theory. We must correct general relativity in a way which prevents superluminal communication.
The existence of timelike loops in, e.g., the Gödel rotating universe, is evidence against general relativity. If timelike loops exist with gravitational waves, that is strong evidence against general relativity.
Our own Minkowski & newtonian model of gravity probably does not allow superluminal communication in a gravitational wave. We will analyze it in the next blog post.
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