The lagrangian LM is of the form
T - V
= kinetic energy - potential energy.
Then a particle which we move with a rogue variation (see the blog post May 21, 2024 for the definition of a rogue variation) would gain kinetic energy in the movement, and the Einstein-Hilbert action S would get a larger value when we try to apply a rogue variation. This is a step toward using "canonical coordinates" in general relativity. We fix the spatial coordinates to freely falling observers.
Picking the freely falling observers is problematic
Our best bet is to pick observers which at some "moment" in the entire spacetime are "static" relative to each other. But this is not possible. Close to a fast moving very heavy neutron star, no observer can be static relative to other observers far away.
• •
Q₁ Q₂
What about picking arbitrary freely falling observers? Consider a system where gravity is negligible and in which we have electric charges. A freely falling observer is anyone with a constant speed. Assume that we determine the kinetic energy T of a charge Q from the measurement made by the closest observer. Then T can vary wildly, depending on which observer happens to be close to Q. This is not going to work.
The solution for the electric charge system is to use observers which are static in some coordinate system of Minkowski space. That is equivalent to using canonical coordinates.
If gravity is not negligible, then the orbits of initially static free-falling observers will eventually cross each other. Which observer will we then use to measure the kinetic energy of a nearby particle? The solution is not well defined.
Using the rogue variation to move the freely falling observers, too
There is a much more fundamental problem in using freely falling observers. Let us use a rogue variation to manipulate the metric g around a particle P and a freely falling observer O close to P.
We can manipulate the metric g in such a way that also the freely falling observer O moves along with P. We can keep the metric g essentially unchanged at P and O, and manipulate it at some distance away.
Then O will not notice any change in the kinetic energy of P, even though proper distances of P from other particles change in the rogue variation.
The solution to this problem would be to introduce canonical coordinates, but that is against the fundamental principles of general relativity.
The definition of kinetic energy in general relativity
Let us have a particle of a mass m whose 4-velocity in the given coordinates is
where τ is the proper time of the particle. Let g be the metric. We define:
Let us have an observer close to the particle, such that the 4-velocity of the observer is uobs. Then the total energy of the particle, as measured by the observer, is
The kinetic energy measured by the observer is defined as:
Ekin = E - m c².
Let us check that the formula above agrees with what the observer should measure intuitively.
We assume that the observer sits still at a fixed spatial coordinates. We assume that the coordinate time x⁰, or t, runs at the same speed as the proper time of the observer. Then g₀₀ = -c².
The only non-zero component of uobs is u⁰obs, and its value is 1.
We assume that g has no off-diagonal components. The total energy of the particle, as measured by the observer, is
E = -p₀ u⁰obs
= -m g₀₀ u⁰
= m c² * dx⁰ / dτ
= m c² * dt / dτ
= m c² * 1 / sqrt(1 - v² / c²),
where v is the velocity of the particle as measured by the observer. The total energy agrees with the formula on special relativity. That is, the formula gives the value which the observer intuitively should measure.
Suppose then that the coordinate time t runs C times as fast as the proper time of the observer. Then u⁰obs is C-fold, dx⁰ / dτ is C-fold, and g₀₀ is 1 / C² -fold. The value of E does not change. It is still the same as what the observer intuitively should measure.
The definition of a rogue variation
Let us recapitulate what a rogue variation is. We calculate the Einstein-Hilbert action integral S from a physical system which consists of particles.
A rogue variation aims to change the path of one of the particles, without the Einstein-Hilbert action noticing any change in the kinetic energy of the particle, or in the action integral S.
Let us denote the coordinates by C, and "canonical coordinates", which are determined by proper distances, by Cc.
The metric is represented by a function g on the coordinates C. Let the representation of the metric on the coordinates Cc be gc.
We vary the system by modifying the spatial distances determined by the metric representation g in such a way that, effectively, the spatial coordinate lines of C move relative to the canonical coordinates Cc.
We keep the spacetime geometry of the system constant. That is, the metric representation gc relative to the canonical coordinates Cc stays unchanged. Note that even if the particle's path in the canonical coordinates Cc changes, its personal gravity field stays in the original path! The particle is "decoupled" from its own gravity field.
The path of each particle stays the same in the coordinates C.
| |
| \
| |
| / <--- "detour"
| |
• P₁ • P₂
m
^ t
|
------> x
Above we have two particles which have a coordinate velocity 0 relative to the coordinates C. The coordinates C were originally identical to the canonical coordinates Cc. Then we modified g in such a way that the particle P₂ did a "detour", if measured in the canonical coordinates.
The metric representation gc does not change. Thus, the action integral of R stays unchanged.
In the matter lagrangian LM, the kinetic energy remains zero, because P₂ stays at the same position in the coordinates C. An observer standing close to P₂ will make a detour along with P₂ and will think that P₂ does not move.
The "potential energy" of P₂ in LM remains as m c², as measured by an observer standing close to P₂.
We conclude that the Einstein-Hilbert action integral S does not change in the rogue variation.
Conclusions
Using freely falling observers to determine the kinetic energy of a particle P does not help. A rogue variation can move also the nearest observers, and the observers think that the kinetic energy of P did not change.
The solution to the problem is to measure the kinetic energy against canonical coordinates, and that is against the principles of general relativity.
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