UPDATE November 17, 2021: We removed the mention of gauge freedom. We added a section about frame dragging.
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Since gravitational waves are quadrupole, only little energy will escape, compared to dipole waves. There is almost complete destructive interference of the outgoing dipole waves. This complicates the detailed analysis of the process. Let us forget those problems for now.
Does a test mass follow a geodesic?
Let us assume that the metric around the Sun is the Schwarzschild metric. Is there a proof that geodesic orbits in the Schwarzschild metric conserve energy?
Yes. The total energy and the specific angular momentum are constants of motion.
Earth loses its total energy in gravitational waves at a rate of 200 W. Thus, the precise orbit of Earth is not a geodesic of the Schwarzschild metric.
Could it be that the orbit is a geodesic in the metric which includes the bent gravity field of Earth?
Is there an analogue of a metric in electromagnetism? Probably not
In an earlier blog post we had the thought experiment in which all particles have the same ratio
q / m,
where q is the (negative) charge of the particle, and m is its mass. Electrons satisfy this condition.
Can we define a "metric" in electromagnetism which would describe the orbits of such particles?
The negative potential of gravity in the metric of general relativity mainly shows up as slowing down of time, or a redshift.
It sounds strange if we have to manipulate the flow of time in an electromagnetic metric.
However, the speed of light does slow down in a polarizable material.
Also, when a negative charge is close to another charge, it has acquired more inertial mass in the interaction. It moves slower than we would expect. This sounds like slowing down of time.
If we lower a positron so close to an electron that we could harvest the entire mass-energy of the pair, they annihilate. This sounds like a black hole.
In the case of pair: a single positive charge and a single negative charge, a metric of general relativity has these effects qualitatively right. Inertia increases when the charges are close to each other.
When the charges have the same sign, it is hard to make a metric which is qualitatively right. We cannot speed up time when the two charges are close to each other, since that would lead to paradoxes. Can we get the effect by contracting the radial metric around a charge? Then a static test charge would not feel a force. That does not work.
We in this blog hold the opinion that curvature of spacetime is just an illusion which happens with an attractive force of gravity. There is no need to describe the electric force with a metric - and we neither can see how it could succeed. The electric force simply does not create the illusion of a metric.
The self-force which a gravitational wave imposes on the mass producing it
In our previous blog post we conjectured that the electric self-force on an electron can be calculated in a very simple manner: just measure how much the electric field differs from a spherically symmetric field at some distance r. That is the "self-field" E which explains the self-force with the formula
F = E e.
In gravity we probably can do the same trick.
But can we describe the self-field with a metric in gravity?
self-field E
<-------
• --->
test mass doing circular motion
Let us assume that the mass is a point particle. We want to impose a force on it. The self-field E would resist its movement in a circular orbit. How do we implement the force in a metric? By slowing down time in the direction of the force?
But if we slow down time at some position P, that position will attract mass from every direction. Is that right? Why would the self-field E attract mass from other directions?
\ bent electric field line
\
P • e- ---> acceleration
/
/
The bending of electric lines of force is a one-sided phenomenon. The self-force F tries to pull the electron back to its old position. Let us assume that the electron started the acceleration from the position P. Could it be that the effect of the self-field would be to pull a negative test charge toward P, even if the charge is to the left of P?
We are interested in the force on a negative test charge relative to the state where the field around the electron would be spherically symmetric around its current position.
The far field of the electron still "lags behind" around the position P. It looks like the repulsion on the test charge close to P is larger than in the case where the electron would be static in its current position.
Thus, the self-field E seems to be just to one direction. In the diagram it pulls to the left. Can we describe such a one-way force with a metric?
Slowing down time is a good way to implement a force. Objects are "moving" in time and they steer to that direction where time flows slower. Can we implement the same effect by manipulating the spatial metric? That is hard since the effect of a force depends on the velocity of an object.
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/ \
/ ● grenade
/ /
O cannon
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Imagine a cannon shooting grenades at different speeds at different directions. We can implement their parabolic orbits by making time to flow slower at lower locations. But could we implement the orbits just by tampering with the spatial metric? The orbits cross each other. How could we remap the distances so that the final locations would be correct? That looks very hard.
It may be that one cannot describe dynamic, changing fields with a metric? One must fall back to a description as a force?
Frame dragging
Or maybe frame dragging is the way to implement the self-force? Around a rotating mass inertial frames are dragged along to the rotating movement.
Frame-dragging around a large rotating mass can be explained by a model where the (negative) energy of the gravity field rotates along the mass. If an infinitesimal test mass is close to it, the test mass has the minimum inertia relative to the large rotating mass if it moves along the large mass. A mechanical clock ticks the fastest if it comoves with the large mass.
Gravitational waves and the flow of time
Our Minkowski & newtonian model suggests that any interacting object gains inertia. A gravitational wave is interacting with the clocks, which can be seen from the fact that the distances which we measure with a laser change as the wave passes.
If mechanical clock parts gain inertia, then the clock will run slower. Also, light will propagate slower.
Linearization of gravity is somewhat suspicious, because in electromagnetism we have charges of both signs, but in gravity we only have positive charges.
Imagine a drum skin. Time runs slower in depressions of the skin. In gravity, depressions of the skin are allowed but hills not. How do we make waves in such a skin?
In the Minkowski & newtonian model gravitational waves are like electromagnetic waves: they make charges to move. Slowing down of time, and apparent changes in distances, is a side effect of the interaction. The metric is just an illusion.
Conclusions
The big question is if we can describe dynamic phenomena, like gravitational waves, with a metric at all.
A metric works when we describe the Schwarzschild field around a spherical mass. It is a static configuration.
Even though Albert Einstein derived gravitational waves from linearized equations in 1916, he continued being unsure if the phenomenon is possible with the full, nonlinear equations.
C. Denson Hill and Pawel Nurowski (2017) write about exact solutions of waves in the full nonlinear theory of general relativity. Andrzej Trautman was able to find solutions. Let us check what they are like.
Plane-fronted waves with parallel propagation (pp-waves) are a description of spacetimes where a plane wave moves. We need to check if these have problems with the metric of time. There is an obvious problem with pp-waves: what kind of a physical process could produce infinite planar waves?
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