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NIKHIL KUMAR JOSHI – ISLAMABAD, PAKISTAN 1-Tel. : 03377648646Q:

Conservation of energy in gravitational fields

What exactly is the meaning of conservation of energy in an accelerated observer in Minkowski spacetime?
This way of thinking to me is the opposite to that of energy conservation in an inertial observer. A typical static observer (say, located at the origin of some coordinate system) in an inertial frame would define the energy-momentum tensor to be
$$T_{\mu
u} = \frac{m}{c^2} u_\mu u_
u \delta^3(x)$$
where $m$ is the energy-momentum of the particle and $u^\mu$ is its 4-velocity.
Now, for a stationary observer in Minkowski spacetime one would have the same energy-momentum tensor but with the energy-momentum of the field. If that stationary observer is accelerated then $m$ would be dependent on $x$ and thus, the definition of $T_{\mu
u}$ would become
$$T_{\mu
u} = \rho u_\mu u_
u\delta^3(x)$$
where $\rho(t, x)$ would be some function of time and space. But then, what exactly is the meaning of energy conservation for the stationary observer (for the particle)?
As a follow up question I would say what is the real meaning of the conservation of the energy-momentum of a field?

A:

Generally, conserved energy-momentum is defined as:
$$

abla_\mu(T^{\mu
u}) = 0
$$
The energy density $\rho$ that is included in $T^{\mu
u}$ is defined by the energy-momentum tensor $T^{\mu
u}$ but it’s not a conserved quantity since it is defined as a space-time distribution and is not a four-vector.
The case you are talking about is not really like what you would usually think of as stationary observers. If the observer is stationary then he will have zero acceleration and there is no need for $T^{\mu
u

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