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The claim that the total energy in the universe is zero can be rigorously justified.

To answer your specific questions:

1. General Relativity is required. It does not apply for Newtonian gravity.

2. It has to be assumed that classical general relativity, with or without cosmological constant, is correct and that the universe is spatially homogeneous on sufficiently large scales. If the universe is infinite the total energy is not really defined, but it is still true that the total energy in an expanding volume of space is asymptotically zero when the region is large enough for the homogeneity of the universe to be a good enough approximation.

3. Here is a link to a paper as requested.

The claim that the total energy in the universe is zero can be rigorously justified.

To answer your specific questions:

1. General Relativity is required. It does not apply for Newtonian gravity.

2. It has to be assumed that classical general relativity, with or without cosmological constant, is correct and that the universe is spatially homogeneous on sufficiently large scales. If the universe is infinite the total energy is not really defined, but it is still true that the total energy in an expanding volume of space is asymptotically zero when the region is large enough for the homogeneity of the universe to be a good enough approximation. The total energy is also exactly zero in a closed spatially finite cosmology whether or not it is homogeneous.

3. Here is a link to a paper as requested.

The claim that the total energy in the universe is zero can be rigorously justified.

To answer your specific questions:

1. General Relativity is required. It does not apply for Newtonian gravity.

2. It has to be assumed that classical general relativity, with or without cosmological constant, is correct and that the universe is spatially homogeneous on sufficiently large scales. If the universe is infinite the total energy is not really defined, but it is still true that the total energy in an expanding volume of space is asymptotically zero when the region is large enough for the homogeneity of the universe to be a good enough approximation.

3. Here is a link to a paper as requested.

The claim that the total energy in the universe is zero can be rigorously justified.

To answer your specific questions:

1. General Relativity is required. It does not apply for Newtonian gravity.

2. It has to be assumed that classical general relativity, with or without cosmological constant, is correct and that the universe is spatially homogeneous on sufficiently large scales. If the universe is infinite the total energy is not really defined, but it is still true that the total energy in an expanding volume of space is asymptotically zero when the region is large enough for the homogeneity of the universe to be a good enough approximation.

3. Here is a link to a paper as requested.

The claim that the total energy in the universe is zero can be rigorously justified.

To answer your specific questions:

(1) General Relaitivty is required. It does not apply for Newtonian gravity

(2) It has to be assumed that classical general relativity with or without cosmological constant is correct and that the universe is spatially homogeneous on sufficiently large scales. If the universe is infinite the total energy is not really defined but it is still true that the total energy in an expanding volume of space is asymptotically zero when the region is large enough for the homogeneity of the universe to be a good enough approximation

(3) Here is a link to a paper as requested.

The claim that the total energy in the universe is zero can be rigorously justified.

To answer your specific questions:

1. General Relativity is required. It does not apply for Newtonian gravity.

2. It has to be assumed that classical general relativity, with or without cosmological constant, is correct and that the universe is spatially homogeneous on sufficiently large scales. If the universe is infinite the total energy is not really defined, but it is still true that the total energy in an expanding volume of space is asymptotically zero when the region is large enough for the homogeneity of the universe to be a good enough approximation.

3. Here is a link to a paper as requested.