ADM have proved that in an asymptotically Minkowski spacetime, the ADM mass and momentum are conserved.
Let us check how ADM handle a few examples which seem to defy conservation of momentum.
An almost massless rigid vessel with pressure
Let us make a vessel of almost massless, extremely rigid material. Let us push almost massless extremely rigid spheres into the vessel.
When the vessel is full in the euclidean geometry, we keep pushing more spheres in. The space inside the vessel stretches under the pressure and we can keep inserting more spheres.
Since we did a lot of work, the ADM or Komar mass of the system is positive.
Let us then pull the vessel and each object inside it with massless tethers. Since each object which we pull is almost massless, we can make the system to accelerate very fast.
The system seems to have an almost zero inertial mass though it contains significant mass-energy. This breaks conservation of momentum as well as several other laws.
Two very massive weights close to each other
In an earlier blog post we showed that a 1 kilogram weight on the surface of a neutron star seems to have < 1 kg of gravitational mass, but > 1 kg inertial mass if pulled with a very rigid tether from far away.
Suppose that we have two very heavy weights close to each other. For example, it could be two neutron stars. If we first pull one with an almost rigid tether and then the other, is the combined inertial mass more than the combined masses of the weights, even though the gravitational mass is less?
Pulling a massive object with a tether
Suppose that an almost rigid tether is attached to a neutron star. A far-away observer starts to pull on the tether. What does an observer on the surface measure?
The Einstein-Hilbert action and the pressured vessel
Probably the easiest way to analyze these problems is to study how we can minimize the integral in the Einstein-Hilbert action.
https://arxiv.org/pdf/1010.5557
Hans C. Ohanian has a paper about the gravitational and inertial mass in general relativity.
Let us think about the vessel experiment which we described above, based on the action integral.
We may assume that the vessel and the spheres in it are electrically charged. We use an electric field to accelerate the system.
______
| + + |
|_____| -----> E
charged pressured electric field
vessel
Let us first think of the experiment ignoring gravitation. The action is basically the time integral over the kinetic energy T minus the potential energy V:
S = ∫ T - V dt.
time
If we now assume that the vessel and its content are made of massless material, the kinetic energy T = 0, and the vessel would accelerate infinitely fast, unless the gravitational field of the pressure somehow restricts the acceleration.
The integrand in the Einstein-Hilbert action is
1 / (2κ) R + L_M.
In our example, L_M is -V. The role of the kinetic energy T must be played by the increased integral over the Ricci curvature R.
Hypothesis: in a static system, the integral over R can be considered the "energy of spacetime deformation". If the pit in the spacetime moves, then the integral over R is larger, and the growth in the action integral over R can be considered the "kinetic energy of the spacetime deformation."
The relativistic action of a moving point particle is a time integral of
m / γ(v),
where v is the speed of the particle and
γ(v) = 1 / sqrt(1 - v^2 / c^2).
But if we have a moving force field, how do we measure its speed? How do we measure the speed of a moving pit in the metric of spacetime? What is the "speed" of the R term in the action integral?
Let us think of the action integral in a 1+1-dimensional Minkowski space, where we do not use any corrective term γ(v) in the integration.
If we have a cloud of matter moving at a slow speed v, then its contribution is
-mc^2 sqrt(1 - v^2 / c^2)
= -mc^2 + 1/2 m v^2
times the global elapsed time in the integration slice.
How do we measure the speed of R, so that we can define the action integral? If R is caused by a static system of particles, then we can use the speed of those particles. What about dynamic systems?
https://en.wikipedia.org/wiki/Stress–energy_tensor
There may be a sign error in Wikipedia about the pressure in the stress-energy tensor.
Let us work in global cartesian coordinates for t and spatial dimensions. The metric g_ij tells us the distances in local coordinates.
Let us keep a physical system, springs, particles, whatever, in fixed positions in global coordinates. In the Wikipedia article about the Einstein-Hilbert action we vary the inverse metric g^ij and study its effect on the action integral. If we did not make a sign error, then in the usual case, T_00 is the mass-energy density and T_11, T_22, T_33 are -P, where P is the pressure.
The Wikipedia link gives T^11 = P, and the formula there gives T_11 = g_11 * g_11 * T^11 = P, if the diagonal elements in the metric are zero.
Hans C. Ohanian has a paper about the gravitational and inertial mass in general relativity.
Let us think about the vessel experiment which we described above, based on the action integral.
We may assume that the vessel and the spheres in it are electrically charged. We use an electric field to accelerate the system.
______
| + + |
|_____| -----> E
charged pressured electric field
vessel
Let us first think of the experiment ignoring gravitation. The action is basically the time integral over the kinetic energy T minus the potential energy V:
S = ∫ T - V dt.
time
If we now assume that the vessel and its content are made of massless material, the kinetic energy T = 0, and the vessel would accelerate infinitely fast, unless the gravitational field of the pressure somehow restricts the acceleration.
The integrand in the Einstein-Hilbert action is
1 / (2κ) R + L_M.
In our example, L_M is -V. The role of the kinetic energy T must be played by the increased integral over the Ricci curvature R.
Hypothesis: in a static system, the integral over R can be considered the "energy of spacetime deformation". If the pit in the spacetime moves, then the integral over R is larger, and the growth in the action integral over R can be considered the "kinetic energy of the spacetime deformation."
The action integral for fields and R
The relativistic action of a moving point particle is a time integral of
m / γ(v),
where v is the speed of the particle and
γ(v) = 1 / sqrt(1 - v^2 / c^2).
But if we have a moving force field, how do we measure its speed? How do we measure the speed of a moving pit in the metric of spacetime? What is the "speed" of the R term in the action integral?
Let us think of the action integral in a 1+1-dimensional Minkowski space, where we do not use any corrective term γ(v) in the integration.
If we have a cloud of matter moving at a slow speed v, then its contribution is
-mc^2 sqrt(1 - v^2 / c^2)
= -mc^2 + 1/2 m v^2
times the global elapsed time in the integration slice.
How do we measure the speed of R, so that we can define the action integral? If R is caused by a static system of particles, then we can use the speed of those particles. What about dynamic systems?
A sign error for pressure in the Wikipedia stress-energy tensor article?
https://en.wikipedia.org/wiki/Stress–energy_tensor
There may be a sign error in Wikipedia about the pressure in the stress-energy tensor.
Let us work in global cartesian coordinates for t and spatial dimensions. The metric g_ij tells us the distances in local coordinates.
Let us keep a physical system, springs, particles, whatever, in fixed positions in global coordinates. In the Wikipedia article about the Einstein-Hilbert action we vary the inverse metric g^ij and study its effect on the action integral. If we did not make a sign error, then in the usual case, T_00 is the mass-energy density and T_11, T_22, T_33 are -P, where P is the pressure.
The Wikipedia link gives T^11 = P, and the formula there gives T_11 = g_11 * g_11 * T^11 = P, if the diagonal elements in the metric are zero.
The R component in the action is negative energy?
The classical newtonian action is an integral on
T - V
where V contains the mass mc^2 as well as various potential energies, for example, the deformation energy of a spring. Note that energies appear with a negative sign in tye formula.
Let us assume that there are no external forces on the system.
T is the kinetic energy of the mass-energy in V.
But how do we match this to the Einstein-Hilbert action? The lagrangian L_M contains the above ingredients. The term R can be thought as negative energy which reduces the mass-energy in L_M. However, the (negative) kinetic energy term for R is missing.
R is the Ricci scalar which is coordinate-independent, and should not change when we move to a moving frame. Length contraction reduces the integral on R. The reduction can be thought of as negative kinetic energy. But contraction can only reduce R by an amount less than R. On the other hand, the negative kinetic energy can be arbitrarily high at speeds approaching c.
There is an additional problem: since R is a nonlinear result of the positive mass-energies in the system, we cannot determine what is the "speed" of R, if the system is not static in some frame.
SOLUTION: the relativistic lagrangian of mass m is
mc^2 / γ(v),
where v is its velocity and
γ(v) = 1 / sqrt(1 - v^2 / c^2).
It turns out that the length contraction of the integral on R just produces the right formula if we consider the integral as negative energy.
But does the action give the right inertial mass? Suppose that the integral on
R - mc^2
is almost zero but negative. Classically, that would mean that the system has a very low inertial mass. However, such a system does contain a large energy in general relativity. Conservation of momentum is in danger if we use the Einstein-Hilbert action.
SOLUTION: the relativistic lagrangian of mass m is
mc^2 / γ(v),
where v is its velocity and
γ(v) = 1 / sqrt(1 - v^2 / c^2).
It turns out that the length contraction of the integral on R just produces the right formula if we consider the integral as negative energy.
But does the action give the right inertial mass? Suppose that the integral on
R - mc^2
is almost zero but negative. Classically, that would mean that the system has a very low inertial mass. However, such a system does contain a large energy in general relativity. Conservation of momentum is in danger if we use the Einstein-Hilbert action.
The problem goes away if we assume that R is negative? We have been assuming that R is positive inside the mass. In literature, a positive curvature means a convex surface, like a sphere.
Could we in the lagrangian L_M flip the sign? But then the lagrangian probably would not work in finding the right R for M.
It looks like the Einstein-Hilbert action works in finding the right static equilibrium for a system. But it does not work for a moving system because M - R is not the right total energy for the system. We may have a system where M = 0 and R is large. The action claims that the total energy is negative though it is positive.
Hypothesis: a lagrangian can only work if it calculates the total energy of each isolated subsystem right. If we have a body of mass in an asymptotic Minkowski space, then its total gravitational and inertial mass is the ADM mass. Since the Einstein-Hilbert action does not calculate the ADM mass, it cannot work: it will not calculate the right path if two systems interact in an asymptotic Minkowski space.
This is at odds with the fact that many people have proved conservation of momentum in such a case. We need to check the proofs.
The fact that the Einstein-Hilbert action does not calculate the total mass-energy of a system right, has been known from the start. People have defined various pseudotensors in an attempt to determine the total mass-energy of the system.
If we think of a rubber sheet model, we believe that its lagrangian does calculate the mass-energies right. Maybe general relativity should be made more like a rubber sheet?
How to fix the Einstein-Hilbert action?
When a mass M falls into a gravitational pit, it, in a sense, loses some energy because there is a redshift if a particle wants to send its energy to a far-away observer.
Suppose that the mass is M and the redshift is 1%. But the binding energy E is not M * 1%, but rather half of that. The deformation of spacetime probably took the other half of M * 1%.
We may guess that the deformation energy of spacetime is an integral over the Ricci scalar R times some coefficient. The Ricci scalar is the simplest Riemann curvature measure.
The way to fix the Einstein-Hilbert action is to add there a potential term which corrects the total energy to equal the ADM mass or the Komar mass. Pressure does contribute to the total energy of the system, not just in the deformation energy of the matter.
Maybe we should count R as positive energy? Then the simple variation method in Wikipedia, where we are allowed to vary g^00 independently of g^11 does not work. The rubber model suggests that slowing time down by making the pit deeper necessarily causes changes in the metric in a wide area. If we want to vary the metric of the rubber sheet, we cannot change the component g^00, etc., independently of the other components.
Maybe we should count R as positive energy? Then the simple variation method in Wikipedia, where we are allowed to vary g^00 independently of g^11 does not work. The rubber model suggests that slowing time down by making the pit deeper necessarily causes changes in the metric in a wide area. If we want to vary the metric of the rubber sheet, we cannot change the component g^00, etc., independently of the other components.
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