Something that’s been bugging me for a week or two: it was stated in this thread that General Relativity predicts that a static universe cannot exist. Assuming an infinite universe (as was assumed by both Newton and Einstein), why would GR cause that universe to collapse, while Newtonian gravity wouldn’t? And for a finite universe, isn’t F[sub]g[/sub]=Gm[sub]1[/sub]m[sub]2[/sub]/r[sup]2[/sup] – an infintely reaching and perpetually attractive force – as certain to cause collapse as GR? Some math on why GR forbids a static universe would be appreciated. Thanks.
I guess I thought Chronos’s last post in the thread you linked to covered it. Perhaps the missing bit of info is that under newtonian gravity, space is assumed flat. Under GR, a static universe with mass, and without a cosmological constant, space cannot be flat. It has to be curved spatially <insert ubiquitous ballon analogy here>. Oh, one other important piece of information is that the spatial curvature of the universe (i.e. whether the universe is flat, open, or closed in its spatial dimensions) is tied to whether the universe is open or closed in terms of time (whether it expands forever, or collapses back into itself). You can’t pick these separately, they have to match (at least under the assumption of a spatially homogeneous universe). So a spatially closed universe must collapse, which rules out a static universe.
OK, I have a confession to make. The reason I haven’t answered this yet isn’t that I thought I answered it sufficiently before. The reason is that I don’t really know the answer.
In Newtonian physics, if you try to calculate the force on a point mass due to all the other masses in the Universe, you’ll get an indeterminate answer (you can make it anything you want, since there’s an infinite amount of Universe available). Now, Newton looked at this and said “since it’s indeterminate, let’s just say it’s zero”, and justified this by pointing out that the Universe was, after all, static, so that had to be the answer (see Article 10 of the Physicists’ Bill of Rights: We retain the right to justify shaky reasoning, on the grounds that it gives the right answer). In GR, the answer is no longer indeterminate, but I’m not sure exactly why. I think, but this is just a SWAG, that’s it’s due to the finite propogation speed of gravitational effects in GR (they propogate at c, the same speed as electromagnetic effects, as opposed to instantaneously, in Newtonian gravity).
I have always understood that a static solution to Einstein’s equations, even one with an appropriate cosmological constant, is basically unstable. In other words the slightest net velocity or mass energy imbalance causes the universe to either expand or contract. It’s like balancing a pencil on its point - you maybe able to do it for a short while but eventually it’s going to tip over.
None of the three Friedman solutions to GR are static. The closest to static is a verry slow expansion.
In the Newtonian case, isn’t the answer zero by symmetry? If the mass distribution of the universe is homogeneous and isotropic, there’s no reason for the forces to pull in one direction more than any other, so it has to be zero.
I’d always thought this boiled down to the concept of “static” being basically meaningless within general relativity, and not to do with the actual closed, flat, or open shape of the manifold, whatever it might turn out to be. Static relevant to what? By some reference frames, of course static, by others, not remotely. Newtonian physics takes space as just an empty backdrop that billiard balls move around within; relativity takes space-time as something with varying geometry based on one’s reference frame.
Am I mistaken and does “static” actually mean anything (regardless of its truth) in General Relativity?
Yes, “static” can have meaning within the context of GR: You’re looking at the motions of galaxies or other large-scale structures relative to each other.
The Newtonian universe is, indeed, usually taken to be static by symmetry, but it depends on what order you do the sum, and what you consider the origin of your coordinate system. To see this, let’s consider a perfectly uniform finite mass distribution, in the shape of a sphere of radius R (for instance, a planet of uniform density). You’re in a little cave at a distance of r < R from the center. The gravity of everything at a larger radius than you cancels out, so it seems to you just like the planet has a radius of little r. If you keep adding uniform layers to the outside of the planet, nothing changes. Now, though, add layers to the outside extending all the way to infinity. You should still feel like “down” is in the same direction… but now you’re in the center of a uniform infinite mass distribution. The moral of this story is that infinite uniform mass distriburtions are unphysical.
I am in very big trouble after reading following state ment " space doesnot like to remain static it is either expanding or contracting except some special cases" i want to know very eagerly about that some special case
Static solutions exist in GR (the term static has a technical meaning in GR).
In the context of the Friedmann equations, the most obvious case is the degenerate one of Minkwoski spacetime which is just empty (in all senses) universe. As useful as Minkowski spacetime is (it provides the background for special relativity), it’s unrealistic for a general relatiivstic cosmology as the universe is obviously not empty.
There is also one other static case of a FLRW universe known as an Einstein static universe. The Einstein static universe is closed (i.e. it has a hyperspherical geometry) and has a postive cosmological constant. However it too can be dismissed as unphysical as it is unstable (any small deviation, as undoubtedly exists, from the exact solution will result in a universe that is not static).
The special case would be a universe where the attractive gravity is exactly balanced out by the repulsive cosmological constant or whatever the heck you want to call it. It’s a precarious balance, though, like a pencil balanced on its point: Sneeze, and it’ll start expanding or contracting from there.
You are the expert (I am not being sarcastic) and you are entitled to speak with authority on the subject.
I forget over 95% of what I read, but I am certain I remember correctly that numerous Einstein biographers and GR popularizers report that the universe did in fact appear static before Hubble’s discoveries, that there were no static solutions to GR, and that Einstein created the cosmological constant ad hoc to endow the universe with a static quality which GR could not otherwise permit. None that I recall mention the empty space-time solution, although I am less sure of my memory on that point. Therefore your references above to plural static solutions are a great surprise. Do you know of an internet source which might be able to cover that ground in terms comprehensible (if barely) to laymen?
The cosmological constant wasn’t exactly ad hoc. It’s basically a constant of integration. At the time of Einstein, the most reasonable derivation would have found it, but then said “we do not know the value of this constant, and it may well be zero”. It would actually have been a bit overconfident to say that it must be zero, and now, of course, we seem to have found a nonzero value for it.
We may be using the expression “ad hoc” differently.
I meant it in the sense that CC was originally created by Einstein for
the sole purpose of forcing a static universe upon GR, a static universe
being not possible in consequence of the unaltered equations.
I do understand that dark energy provides a modern analogue of the
original CC, even though DE is meant to explain a universe observed
to be accelerating in growth rather than to explain a universe wrongly
believed to be static.
There are static solutions in GR, but under the standard assumptions of cosmology (i.e. homogenity and isotropy), there’s only two* static possibilties Minkoski space and an Einstein static universe (which requires a cosmological constant)
I’m not suprised that Minkowski spacetime was not mentioned as like I said it’s the degenerate case, barely worth mentioning as it cannot be taken seriously as a GR comsological model as the universe clearly does not have a zero energy density.
That said the Milne model of the universe does use a Minkowski spacetime, however the Milne model is inconsistant with the universe both being governed by GR globally and non-empty.
In terms of staticity generally in GR, usually what is required is exact symmetry and/or a pressure (the cosmological constant can be seen as a negative pressure) that exactly cancels out expansion/contraction. A counter example to this though is gravitational collapse, which tends to evolve in to a stationary (a slightly weaker condition than ‘static’) state.
I don’t really know of any websites that may be helpful.
*Sometimes de Sitter Universes are described as static too, but this is only in a very limited sense (they admit static coordinate patches) which doesn’t correspond to the layman’s understanding of static (in the most natural choice of coordinates there is still the usual metric expansion of space).