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I have read a lot about (what is the total energy of universe? And Is it zero?) here, such as, Is the total energy of the universe zero? and other sources. I thought decades about a question, not exactly about: What is the total energy of universe? But: Is the total energy of universe, if has any meaning word "total", absolute or relative?

My question is based on other questions:

  1. Is there meaning for conservation of energy in no inertial frames?

  2. If (1)'s answer is yes, is the total energy of universe absolute for any observer or relative (including observer's own energy!)? So, that if we have two observers, would they sum the same total energy?

  3. If (1)'s answer is no, for a free-falling frame of reference (tiny enough to be considered as inertial), is the total energy of universe absolute or relative for it? Again, if we have two observers, would they sum the same total energy?

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  • $\begingroup$ When you say "relative to the universe" you are implying that there is a frame of reference and observer "outside the universe"? $\endgroup$ – joseph h Sep 26 '20 at 2:52
  • $\begingroup$ I didn't mean to say relative to the universe. I am talking about two different observers inside universe, would they measure same universe enegy? $\endgroup$ – Ahmed Kamal Kassem Sep 26 '20 at 6:34
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From the General Relativity perspective, the energy content of any system is always observer-dependent. In general, different observers measure different energy contents of some system.

About the "zero energy" question, I think this has little physical significance and is more a misconception from interpreting a stress-energy tensor component as the energy density of the universe. Check this answer

About the other questions:

Energy is not conserved in general spacetimes. This only happens when the spacetime possesses a time translation symmetry (roughly speaking, when it "looks the same" independent of the time). An accelerated expanding universe (such as ours) does not have energy conservation. However, in the time scale and space scale of our lives, this energy variation is so small that it looks conserved for any practical purpose.

A few spacetimes, however, possesses more than one "time translation symmetry", which allows two different families of observers to measure conserved energies (it is the case of an uniformly accelerated observer in Minkowski spacetime, for example).

I think energy is generally an absolute concept in General Relativity, rather than a relative one. What I mean here is that the curvature spacetime is related to the mass/energy that it contains, rather than a "difference of masses" (whatever that means).

It should be pointed out that there is no consense of a unique fail-proof definition for the total mass/energy of a spacetime. Maybe the most standard used out there is the "ADM mass" which works great in a large class of scenarios.

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General relativity does not have a conserved, scalar, globally conserved measure of energy that can be defined in all spacetimes. General relativity does have strict local conservation of energy.

The basic reason for this is that energy is not a scalar in relativity, it's one component of the energy-momentum vector. In a curved spacetime, you can't add vectors defined at different points, so that means there is in general no way to even define the total energy in a curved spacetime.

Is the total energy of universe, if has any meaning word "total", absolute or relative?

As you suspected, the total energy of the universe is not defined. Locally, conservation of energy in general relativity is defined in such a way that it is valid in any frame of reference. There is a stress-energy tensor, a 4x4 matrix of numbers, one of whose elements can be interpreted as the density of energy. All the components of the stress-energy tensor are relative, i.e., they have different values when you change frames of reference. However, the conservation of these numbers holds in all frames.

In another answer, Ian wrote:

Energy is not conserved in general spacetimes. This only happens when the spacetime possesses a time translation symmetry (roughly speaking, when it "looks the same" independent of the time). An accelerated expanding universe (such as ours) does not have energy conservation.

This is wrong. Noether's theorem doesn't apply to general relativity. For a spacetime with a time translation symmetry, it's true that one can define a conserved energy for test particles, but that doesn't have anything to do with the question of whether the spacetime as a whole has a conserved energy. In a stationary spacetime, it's trivially true that energy is conserved, because no observables are changing over time. This triviality is different from Noether's theorem, which is not a triviality. Noether's theorem has to do with symmetries of the Lagrangian, not symmetries of the system's state.

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