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The second law as typically formulated says something akin to "the entropy of the entire universe is increasing in time." When passing to general relativity this is immediately suspect, as in that theory is no globally invariant notion of time. Of course we can make sense of a specific arrow of time locally, but the second law is not a local statement– entropy in subsystems can clearly decrease, as long as the entropy of another subsystem somewhere else increases by an equal or greater amount.

Alternatively, we could expect a statement like "for every spatial slicing of spacetime, the global entropy is increasing." This seems like it could be a natural generalization of the second law, however it also seems a fairly strong statement as we have a lot of freedom in how we choose to foliate our spacetime.

Energy conservation also seems like a slippery issue here-- it is clearly needed to get an increasing entropy, and there are some obvious situations where it is violated, such as when the cosmological constant is nonzero. For the purposes of this question I will assume $\Lambda = 0$, although I would like to know more about what other assumptions are needed to get something like conservation of energy in GR.

In summary, my questions are: what is the "correct" generalization of the second law in general relativity, and what assumptions are needed (including things like $\Lambda = 0$) for such a statement to be physically correct?

Qmechanic
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Panopticon
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1 Answers1

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TLDR: It is the time of comoving observers (cosmic time) w.r.t. which the evolution of the global entropy of the Universe should be measured. Also, $\Lambda > 0$ only results in a const offset in both the total energy and total entropy of the observable Universe.

You are right to assume that

Of course we can make sense of a specific arrow of time locally, but the second law is not a local statement

since in relativistic physics, the interval of time is observer-dependent. However, on large scales, there is a certain frame of reference from where the Universe appears homogenous and isotropic. W.r.t. such observers, the Universe came into existence at a definite finite time in the past.

These comoving observers (i.e., moving with the Hubble flow) induce a slicing on spacetime, and their induced time coordinate is called cosmic time. It is their time coordinate w.r.t. which the evolution of the global entropy of the Universe should be measured.

For you second question, i.e.,

Energy conservation also seems like a slippery issue here-- it is clearly needed to get an increasing entropy, and there are some obvious situations where it is violated

let us assume a de-sitter universe, instead of a FLRW (the former is a special case of the latter). Then the metric reads

$$d s^{2}=\left(1-\frac{\Lambda}{3} r^{2}\right) d t^{2}-\frac{1}{1-\frac{\Lambda}{3} r^{2}} d r^{2}-r^{2} d \Omega^{2}$$

This spacetime has a horizon at $r = \ell_\Lambda = \sqrt{3/\Lambda}$ (the coefficient od $dr^2$ diverges). The volume of such a (observable) Universe is $V = 4\pi\ell_\Lambda^3 / 3$, and therefore, the total energy contained in such a Universe is $E = \rho V$, where $$ \rho = \Lambda / 8 \pi G$$ is the energy density of vacuum. As you can see, $E$ is still constant for the observable universe even with $\Lambda > 0$.

The entropy of such a Universe might be stated using the holographic principle as

$$S = A/4l_P^2$$

where $A = \pi \ell_\Lambda^2$ is the area of the horizon and $l_P$ is the Planck length. Again, the entropy (of the de-sitter Universe, therefore excluding any matter) also turns out to be constant in this case.

So, in summary, $\Lambda >0$ still results in a const energy $E$ and also a const entropy $S$. So, one can still talk about the entropy of the matter knowing that a positive $\Lambda$ would only offset these by a const value. Note I assumed a de-sitter Universe, but out Universe at late time would approximately approach such a condition.

S.G
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