Can Spacetime Curvature Exist Independently of Mass?

[Stingray]

So a decaying orbit would still be considered a stable orbit by people who unlike me understand?

Yes, usually. But the meaning of these concepts becomes unclear if the orbit is decaying very quickly.

 

[Chris Hillman]

Geons, anyone?

Hi again, notknowing,

Einsteins field equations are nonlinear. One could interpret this to mean that curvature is itself the source of curvature (thus not only mass). Would it be possible to find a stationary (non-zero) solution of the (non-linearised) field equations without a mass being present - a kind of spacetime deformation which has an existence independent of mass ?

It sounds very much like you want to notion of a "geon", which was introduced by Wheeler under the slogan "mass without mass". Try these: http://arxiv.org/find/gr-qc/1/OR+ti:+geons+ti:+geon/0/1/0/all/0/1
(I should caution that some of these papers make controversial assertions. Note too that there are various ways to define "mass" which may be more or less appropriate in different circumstances.)

I'm pretty sure there is a soln to Einstein's field eqns for a massless universe which contains only energy.

This probably is not what you want (nothing to do with "inflation"--- by the way, I think you meant "preBB cosmology"), but there are many nontrivial exact vacuum solutions which have zero ADM mass. A large family of them was found by Papapetrou (this is a subfamily of the Ernst vacuums, the family of all stationary axisymmetric vacuum solutions to the EFE), but most of these are not asymptotically flat. A simple AF example has been discussed by Bonnor.

The issue of interpretation is often vexed in gtr, and it is not clear that these solutions are valid. In weak-field gtr, the two halves of the Ernst equation decouple, which means that you can treat sources have independently variable mass-energy density and momentum density. That is, in Newtonian gravitation, a spinning disk and a nonspinning disk (with uniform density and the same mass) give the same gravitational field, but in gtr the fields differ (the differences arise from a zero or nonzero gravitomagnetic field). But you would presmably not want to allow a source which has nonzero angular momentum but zero mass! If I recall correctly, the massless AF Bonnor vacuum does have zero Komar mass but nonzero Komar angular momentum. (I am too lazy to check this right now.)

Well, what if there are no quadropole moments?

masudr, I am coming into this rather late, but I guess that you meant: do systems with constant mass, angular momentum, and quadrupole moment but with time varying octupole moments (next higher order after quadrupole moments) produced gravitational radiation, according to the linearized-EFE? The answer is yes. (More pedantically, we can define moments for mass-energy and momentum of all orders, so that in addition to "mass-quadrupole moment" and so on, we can compute a "current-quadrupole moment" which gives another contribution to the radiation, and so on for higher orders.)

So I take it between all the flowery language that the answer is yes orbits in GR are not stable, while the effect is very smalll, orbits spiral towards the center.?

Oh dear, Jennifer, I can't imagine how you might have been led to that conclusion, but be assured that it is not true! I agree with stingray and pervect that you appear to have confused closed or decaying orbits with stable orbits-- that's unfortunate, but probably not your fault (probably you were reading something which didn't bother to use the correct terms, or to note that these are distinct concepts?)

The standard definition of stable, as used in dynamical systems theory and in applications in gtr, e.g. to test particle orbits, basically means that the orbit is stable under small perturbations in the sense that small changes don't change the orbit drastically. It does NOT contradict either quasi-Keplerian motion (so that the orbits are not closed) or inspiral (aka orbital decay) due to gravitational radiation carrying away energy from the system.

Chris Hillman

 

[notknowing]

Hi again, notknowing,



It sounds very much like you want to notion of a "geon", which was introduced by Wheeler under the slogan "mass without mass". Try these: http://arxiv.org/find/gr-qc/1/OR+ti:+geons+ti:+geon/0/1/0/all/0/1
(I should caution that some of these papers make controversial assertions. Note too that there are various ways to define "mass" which may be more or less appropriate in different circumstances.)

Hi, indeed the concept of a "geon" is what I had in mind. I quickly looked through some of the references (on arxiv) you gave and it seems I have a lot of reading to do. I had never heard of a "geon" before, and probably most members in this forum neither (as no one else pointed me to this). It seems to indicate that intuition is worth something in physics .
Form the quick look I had at some of the articles, it seems that they are working with solutions of the linearized equations, but this is not what I had in mind. I think that the fully nonlinear equations allow for much more interesting (soliton-like) solutions, though they are probably very hard to solve. Suppose for a moment that such stable solutions indeed exist, then they could probably be of relevance to the "missing mass" problem.

 

[Chris Hillman]

Vacuum solutions with vanishing ADM mass

I wrote a lengthy comment on this thread the other day, but unfortunately due to system instability at PF I lost my work. So I'll keep this brief, at least until the system becomes more stable.

The original poster may or may not be interested in solutions answering to the description in the title, but Papapetrou discovered a large class of massless stationary axisymmetric vacuum solutions, which is of course a subclass of the Ernst vacuums (the class of all stationary axisymmetric vacuum solutions). The Papapetrou vacuums are in general not asymptotically flat, but Bonnor gave a simple example of a nontrivial asymptotically flat stationary axisymmetric Ernst vacuum with vanishing ADM mass.

The issue immediately arises of what could cause such behavior. That is, what might be the source of such a gravitational field? In particular, what kind of isolated, compact configuration of moving matter might produce an asymptotically flat stationary axisymmetric field with zero mass?

As it happens, in weak-field gtr, the field equations for stationary axisymmetric vacuums reduce to two uncoupled equations whose source is respectively mass-energy and (angular) momentum.

Imagine an isolated thin uniform density disk. In Newtonian gravitation, because of the axial symmetry of the source, a spining or nonspinning disk will produce identical gravitational fields. In gtr, spinning up the disk so that the rim is moving at nonrelativistic velocities will in principle produce a small gravitomagnetic field, so in gtr, in principle a physicist orbiting a massive disk can in principle tell whether or not it is spinning by testing within his spaceship/laboratory for possible gravitatomagnetic effects on test particle motion.

(Non-relativistic: spinning up the disk will eventually increase the effective mass-energy density inside the disk. The additional energy has of course come from whatever physical process we are using to spin up the disk--- which we profoundly desire to avoid modeling! This last brings up the difficulty of devising thought experiments, due to the fact that all forms of mass-energy gravitate, which can create problems when you wish to make something change without modeling exactly how the necessary energy is provided. Compare say Born's thought experiment in which some non-EM force is used to uniformly accelerate two identical charged particles away from each other. If you try to construct the analogous gravitational experiment, you are liable to wind up postulating infinitely strong yet massless strings pulling on the two particles! But such massless strings are sufficiently unphysical to possibly destroy the value of the idealization.)

A physically reasonable source presumably shouldn't have angular momentum without mass. But at least in weak-field gtr, it seems that the field equation does not rule out this possibility. Such overgenerosity is perhaps analogous to the fact that neither Newton's field equation (the Laplace equation, in the classical field theory reformulation of Newtonian gravitation) nor the EFE rule out sources which possesses negative mass.

Anyway, my point is that these considerations suggest that these solutions may be artifacts of using inappropriate boundary conditions.

Chris Hillman