General relativity has problems defining the kinetic energy in matter lagrangians

Let us consider the Einstein-Hilbert action once again.

It contains the ordinary matter lagrangian density L_M. The ordinary matter lagrangian is typically of a form

       L = V - T,

where V is the potential energy and T is the kinetic energy. But how do we determine the kinetic energy T? If we would be working in Minkowski space, then we would measure velocities against fixed inertial coordinates.

                   A                         B

                   ● +                      ● +

                   |---------------------------------> x

                   0       1       2       3

Let us have two electric charges A and B which repel each other. In fixed Minkowski coordinates, a lagrangian density

       L = V - T

calculates their paths right.

Let us then use such coordinates that a small patch of the spatial coordinates moves along both charges: their kinetic energy T stays zero in these coordinates. The coordinates move, but the metric stays as the fixed Minkowski metric. In the diagram, we let the metric to change in such a way that the distance between x = 1 and x = 2 grows.

General relativity allows us to use any coordinates, as long as the spatial coordinates do not move superluminally.

In the new coordinates, the matter lagrangian calculates the paths incorrectly.

How can we fix this? Should we transform the matter lagrangian somehow when we move to new coordinates?

A tensor is a physical parameter which can be "easily" converted between various coordinate systems. The conversion is local. If we only manipulate the metric between x = 1 and x = 2, then a tensor describing locally the charge A or B does not get transformed.

There does not exist a fixed Minkowski coordinate system in general relativity. In our March 17, 2023 blog post we remarked that one cannot use static coordinates at all for a fast moving neutron star because the star moves superluminally relative to static coordinates.

In literature, when working with general relativity, people assume that the sole long-distance force is gravity. Other forces only act locally. Does this save us from problems? One can then use a locally defined, almost Minkowski coordinate system. This helps to alleviate the problem. However, then it should be explicitly stated in textbooks that the coordinates must imitate Minkowski coordinates - one cannot choose the coordinates freely.

Suppose then that A is in faraway space and B in on the surface of a very fast moving neutron star. We cannot use a static coordinate system. When we determine the kinetic energy of B, in which coordinate system we should do it?

In our own Minkowski & newtonian gravity, the coordinate system is always the fixed Minkowski coordinates. Our own gravity model does not suffer from the exact problem described here. However, we do have a problem defining the kinetic energy when we have a complex system of interacting fields. What is the amount of energy and where is it located?

Conclusions

We have not seen in literature any mention about the problems of defining kinetic energy in general relativity. If we cannot define kinetic energy, we cannot define the stress-energy tensor. The problem is theoretically fundamental - though not relevant in practical applications.