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My question is the following. Are there solutions to the Einstein field equations, which have the property that there is a hypersurface of constant time and to the past of that surface space is empty (Minkowski space-time) and to the future it is not (non-vanishing stress-energy tensor)? At first thought it seems strange to have nothing and suddenly something. On the other hand define $g_{\mu\nu}(x)$ to be the Minkowski metric for the past of the surface and any functions (sufficiently smooth) that match it on the surface and are different than the Minkowski for the future. There are many ways (infinitely many) to do this. Then one can define the stress-energy tensor using Einstein's equations. This would seem to work, but there are certain conditions on the tensor, for example positivity, or other restrictions for physical reasons that I am unaware of, so it may be there are no such functions.

So the questions is: does anyone know of an explicit example or a more convincing argument than "there are so many functions, there has got to be some that work"? Of course a reason why it doesn't work, if that is the case, would be good too.


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5 Answers

up vote 6 down vote accepted

I believe that no such spacetime exists, if the matter is assumed to satisfy an inequality known as the dominant energy condition. The dominant energy condition says that, if $\xi$ is a future-directed timelike vector, then $-{T^a}_b\xi^b$ is a future-directed timelike or null vector (sign conventions, etc., following Wald's book General Relativity). Heuristically, this condition means that an observer at any location will always measure an energy-momentum 4-current in his vicinity that is flowing at less than or equal to the speed of light.

With this condition, one can show that, if space is empty at one time (i.e., if there is a spacelike Cauchy hypersurface along which $T=0$), then it vanishes at all times (Wald, p. 219). If in addition spacetime is Minkowski along some Cauchy surface, then the initial value theorems say that it's Minkowski at all times (Wald, Chapter 10).

Heuristically, the above argument says that, if there's no matter at one time, but there is at a later time, then matter must have popped out of nowhere. The dominant energy condition doesn't allow that.

I don't know whether the result holds if you assume something weaker than the dominant energy condition (e.g., the aptly-named weak energy condition). Certainly you need some sort of restriction on the allowed properties of the matter, as Lubos Motl says: if any $T_{ab}$ is allowed, then any smooth metric is allowed.

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Thanks, that is an answer. I'll check out the details in Wald's book. –  MBN Jan 27 '11 at 21:48
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Dear MBN, if you allow the future metric to be sourced by an arbitrary $T_{\mu\nu}$ that doesn't have to be derived from any actual type of matter or fields, then your spliced spacetime may be constructed. However, it's simply because Einstein's equations become tautologies if you allow $T_{\mu\nu}$ to be anything. For any geometry (metric), you may find a $T_{\mu\nu}$ profile such that Einstein's equations are satisfied - just calculate the "right" stress-energy tensor from Einstein's equations.

Just take the Minkowski space for $t<0$ and extend the metric tensor components to arbitrary functions of $t$ that are infinitely differentiable, even at $t=0$, but that are non-constant for positive $t$. For example, you may take such functions to be $$g_{\mu\nu}(t) = g_{\mu\nu}(t<0) + C_{\mu\nu} \exp(-1/t^2)$$ for positive $t$.

Now, calculate the curvature and Einstein's tensor out of this arbitrary metric, and you will know what $T_{\mu\nu}$ should be declared to be the source of this gravitational field. However, you will never derive such a "suddenly turning on" matter source out of any well-defined equations. It's because you would face a similar problem e.g. for the electromagnetic field that could source the strange spliced gravitational field.

However, the electromagnetic source also can't be turned on "suddenly", unless it's sourced by an electric charge distribution $j_\mu$ that also has to be turned on suddenly. So something has to be "externally inserted" to any physical system to achieve the change of the behavior in $t=0$.

What's more important is that you can't get your spliced spacetime out of the vacuum Einstein's equations, i.e. those with $T_{\mu\nu}=0$. Einstein's equations are second-order partial differential equations for the metric components $g_{\mu\nu}(t,x,y,z)$. Throw away all the indices and useless dimensions and solve a morally similar problem, the equation $$d^2 x/dt^2 = 0$$ Clearly, this equation has a unique solution for given initial conditions. Well, in this particular form, the solutions have to be linear functions, but more generally, the solution is uniquely determined by the initial conditions. The same can be proved for the partial differential equations from a broad class - including Einstein's equations.

To summarize, you can say that the change of the behavior can only be achieved if some external sources of the fields are manually added at $t=0$. Of course, if you gradually turn them on, the response will be gradually increasing, too. But there is nothing mysterious about it. If you suddenly "buy" some matter from another spacetime and insert it to Einstein's equations starting from $t=0$, e.g. by an infinitely smooth function, the metric starts to get curved in a similar way after $t=0$, too.

Because the stress-energy tensor has to have a vanishing covariant divergence (because the Einstein's tensor has the same property identically), it will constrain the kinds of stress-energy tensors that may be turned in this way. It's likely that the required stress-energy tensor - e.g. one calculate from the metric I wrote in a displayed equation above - always violates some (or all) energy conditions right after $t=0$. In particular, the components of the required $T_{\mu\nu}$ will be "mostly spacelike" which is forbidden by the null or dominant energy conditions.

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Thanks for the quick reply Lubos (is it pronounced Lyubosh?), but this is not really an answer. You are just rephrasing and expanding my question. Which is not a bad thing, if my writing was not so clear. –  MBN Jan 27 '11 at 21:51
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Are there solutions to the Einstein field equations, which have the property that there is a hypersurface of constant time and to the past of that surface space is empty (Minkowski space-time) and to the future it is not (non-vanishing stress-energy tensor)?

This is essentially what happens in inflationary theory. Though, not quite in the strong mathematical sense that you are seeking. There one starts out with a scalar field in an expanding background metric, usually deSitter. What happens is that the exponential expansion of spacetime amplifies quantum fluctuations in the scalar so that the end of the inflationary phase we are left with a scalar with a scale-invariant spectrum of perturbations. It is these perturbations that go on to see structure formation in the early Universe.

Einstein's equations are satisfied by the initial scalar field + deSitter system. Of course these considerations involve QFT on a curved background so one has to go beyond the framework of classical GR in calculating the spectrum of perturbations.

You can read more about this in Chapter 8 of Mukhanov's excellent book "Physical Foundations of Cosmology".

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Just to be clear, it is certainly not true in inflationary theories that spacetime at early times is assumed to be Minkowski as in this question. –  Ted Bunn Jan 28 '11 at 15:01
It is not exactly what I was asking but is interesting on its own. –  MBN Jan 28 '11 at 15:50
@Ted I did mention the fact that the spacetime is deSitter in inflationary scenarios. –  user346 Jan 28 '11 at 16:16
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Interestingly, there is still a hypothetical possibility that the opposite might have been the case, i.e. non-empty first and empty then, pending the results of the AEgIS experiment on gravitational behavior of antimatter:


If it turns out that matter and antimatter repel each other gravitationally, it will lend support to the theory by Chardin and Benoit-Lévy that the universe may be at large scales an emulsion of matter and antimatter that cancel each out gravitationally, so that the universe is "empty" at large scales (meaning the net gravitational density of large volumes is zero):


In this model, the universe might have started with big enough energy density (radiation era) to have closed geometry and finite size and content, and through the matter-antimatter formation and separation process the net energy density (from a gravitational viewpoint, i.e. Rho in Friedmann equations) might have progressively vanished, leading to an open geometry still with finite size and matter content.

Just a possibility to keep in mind until the jury is out on AEgIS. BTW, I perceive that some physicists are actually afraid that it finds out that matter and antimatter repel each other, while in contrast some very good wine would be uncorked in France.

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@Anna V: the Chardin & Benoit-Lévy hypothesis implies an entirely different interpretative framework of the CMB. They have just shown that their theory predicts the first peak at more or less where it is. Clearly the amount of man-hours that they have dedicated to the topic is nothing compared to what has been globally invested in the CMB interpretative framework based on the Lambda-CDM model. My guess is that they are waiting for the results of the Aegis experiment before going further, which I see as sensible. –  Alex Feb 19 '11 at 15:10
thanks for the clarification –  anna v Feb 19 '11 at 15:26
back in 1994 they measured the effect of the tides on the beam energy of LEP cdsweb.cern.ch/record/260368/files/CM-P00061203.pdf . They attribute the changes to the deformation of the ground due to the tides. It seems nobody looked for a difference between electrons and positrons over the thirty kilometer beam rotations over hours. –  anna v Feb 19 '11 at 15:41
Or protons and antiprotons at the tevatron. –  anna v Feb 19 '11 at 15:46
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If there existed at some point the same amount of matter and antimatter, should there not be a signature in the microwave background radiation? I would expect that the annihilation of electrons and positrons should still be a separate bump?

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Sorry, this should have been a comment to the question above. It addresses a question coming from the Aegis experiment . –  anna v Feb 19 '11 at 14:53
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