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I am slightly confused with the invariance of four-volume element. The orthodox way to show it is to prove that Jacobian is one, that I did, however in many textbooks I find a reasoning that because we have Lorentz contraction on one hand and time dilation on the other hand, the product is invariant then. I can not agree with that:

The problem can be stated like this: one has a cube of sides [itex] \Delta x [/itex] which lives only for time [itex] \Delta t [/itex]. What are the sides and life of the cube in moving reference frame?

I think that solution is like this: in moving reference frame there will be a length contraction but time also will seem to shorter [itex] ds = dt \sqrt{1-v^2} [/itex] so the product is not invariant.

Am i wrong? would appreciate any answer, spent lots of time on thinking about it.

The problem can be stated like this: one has a cube of sides [itex] \Delta x [/itex] which lives only for time [itex] \Delta t [/itex]. What are the sides and life of the cube in moving reference frame?

I think that solution is like this: in moving reference frame there will be a length contraction but time also will seem to shorter [itex] ds = dt \sqrt{1-v^2} [/itex] so the product is not invariant.

Am i wrong? would appreciate any answer, spent lots of time on thinking about it.

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