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How can one show from General Relativity that gravity is attractive force, and under which conditions it becomes repulsive, also why positive energy vacuum drives repulsive gravity?

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it seems like there might be an actual question in this, but right now I don't quite see it... I don't quite understand what it is asking... – Benjamin Horowitz Aug 6 '11 at 15:31
It seems like a perfectly reasonable question to me. – Ben Crowell Aug 6 '11 at 16:28
Possibly related: – Qmechanic Aug 6 '11 at 18:15
I think that's an excellent question. And retagged... – Stan Liou Aug 12 '11 at 11:47
other recent viewpoint: CPT symmetry and antimatter gravity in general relativity "... Starting from the CPT invariance of physical laws, we transform matter into antimatter in the equations of both electrodynamics and gravitation...". – Helder Velez Aug 14 '11 at 21:22

The Einstein field equations actually don't say anything at all about the nature of matter. Their structure is that they relate a certain measure of spacetime curvature G to the stress-energy tensor T: $G_{ab}=8\pi T_{ab}$. The stress-energy tensor describes any matter that is present; it's zero in a vacuum. Trivially, you can write down any equations you like describing an arbitrary spacetime that you've made up, and then by calculating G you can find the T that is required in order to allow the existence of that spacetime. However, that T may have properties that are different from those of any known type of matter. T has a very specific structure for certain types of matter, such as electromagnetic radiation or "dust" (meaning a perfect fluid made of particles that have velocities $\ll c$ relative to one another). There are various conjectures, called energy conditions, about what types of stress-energy tensors are physically possible for realistic types of matter. They have names like the weak energy condition (WEC), strong energy condition (SEC), etc. The WEC amounts to a statement that the energy density is never negative in any frame of reference. If it was violated, then you could get repulsive gravity. Basically all energy conditions are known to be violated under some circumstances. Here is a nice discussion of that:

The cosmological constant is sometimes taken as a separate term in the Einstein field equations, but it can also be treated as a type of matter with a certain contribution to the stress-energy tensor. A spacetime with only a cosmological constant, and nothing else in it, violates various energy conditions.

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very interesting paper, i tend to believe that these (energy) conditions are (will be eventually shown to be) only valid in the thermodynamic limit. But so far there is no evidence that violating "matter" is nothing more than a constant (until some inhomogeinities in the expansion acceleration rate are observed) – lurscher Aug 8 '11 at 15:09
i need some time to look through paper you mentioned...As far as we know today accelerated expansion of universe might be driven by positive cosmological constant (energy density of vacuum). I just wondered if there is some simple physical explanation or proof that gravity is always "attractive", except in some simple cases. Now I see that explanation is probably not trivial. But thank you for introducing me to energy conditions, and answer that is joy to read. – Newman Aug 9 '11 at 1:02

We need a clear operational meaning about what it means for gravity to be "repulsive". If we think of it too naively... say a distant observer in Schwarzschild spacetime viewing a particle orbit, for whom Schwarzschild time $t$ has a clear operational meaning and Schwarzschild radial coordinate $r$ is good enough. Can the orbit have positive $d^2r/dt^2$, i.e., have outward rather than inward acceleration? Yes, absolutely: in fact, it must be the case, because in those coordinates a radially freefalling particle never reaches the horizon.

But that's just perverse. Let's look for a local operational definition. For example, take a small (near each other) collection of comoving test particles, four-velocity $u$, and see what they do. Their geodesic deviation is given by the Riemann curvature tensor, and if we have a small ball of volume $V$, then $$\lim_{V\to 0}\frac{\ddot{V}}{V}\vert_{t=0} = -R^\alpha{}_{\mu\alpha\nu} u^\mu u^\nu,$$ and this contracted Riemann tensor is the Ricci curvature $R_{\mu\nu}$. Thus, the attractive or repulsive behavior of the ball of test particles is given by the Ricci curvature.

Take a gander at the strong energy condition: for every future-pointing timelike vector $u$, $$\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)u^\mu u^\nu \geq 0.$$ It's not immediately obvious as to what this actually means, but contracting the Einstein field equation $R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8\pi T_{\mu\nu}$ gets you $R - \frac{1}{2}Rg^\mu{}_\mu = 8\pi T$, i.e., $R = -8\pi T$: the Ricci scalar is just the negative trace of the stress-energy tensor, up to a constant. Back-substituting into the field equation, and voila: $$R_{\mu\nu} = 8\pi\left(T_{\mu\nu} - \frac{1}{2}Tg_{\mu\nu}\right),$$ so the SEC just says that $R_{\mu\nu}u^\mu u^\nu \geq 0$.

Conclusion: for this reasonable local notion of what it means for gravity to be attractive/repulsive, gravity is non-repulsive if, and only if, the strong energy condition holds.

In the case of a perfect fluid, $T^{\mu\nu} = (\rho+p)u^\mu u^\nu + pg^{\mu\nu}$, so that $T = T^\mu{}_\mu = -\rho + 3p$. From the frame comoving with the fluid, substitution gives $\rho + 3p\geq 0$. On the other hand, approaching a lightlike $u$ in one of the spatial directions gives $\rho + p\geq 0$, this step being justified by continuity.

Since the Einstein field equation with a cosmological constant just adds a $\Lambda g_{\mu\nu}$ opposite the stress-energy tensor, it is equivalent to a perfect fluid with density $\rho = \Lambda/8\pi$ and pressure $p = -\Lambda/8\pi$. Thus, we need positive vacuum energy density to break the SEC.

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Stan Liou, I enjoy reading your answer very much. It seems convincing to me. Thank you. – Newman Aug 14 '11 at 1:08

In the context of General Relativity, small test particles move on geodesics. A geodesic is a generalization of a straight line (for example, on a curved surface like a football). The geodesics are determined by the metric $g_{\mu \nu}$.

The Einstein equation is $G_{\mu \nu} = 8 \pi T_{\mu \nu}$.

$G_{\mu \nu}$ consists of the metric $g_{\mu \nu}$ and its derivatives.

$T_{\mu \nu}$ is matter, and has specific forms depending on what matter is present, like dust, radiation, or lambda. You can think of $T_{\mu \nu}$ as the input to the problem, like a heavy star sitting at the center of empty space.

Due to the concrete form of $G_{\mu \nu}$ as written down by Einstein, when you solve the equation for dust or radiation, $g_{\mu \nu}$ will be such that test particles will appear to be attracted to the matter described by $T_{\mu \nu}$. For a heavy star sitting at the center of empty space, the geodesics of test particles coming from very far away will bend toward the heavy star. This solution is actually famous for being the first explicit solution of a problem in the framework of General Relativity, and is called the Schwarzschild solution or Schwarzschild metric after Karl Schwarzschild.

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"Due to the concrete form of Gμν as written down by Einstein, when you solve the equation for realistic forms of matter, gμν will be such that test particles will appear to be attracted to the matter described by Tμν." This isn't true. You can write down a metric describing repulsion, and from it calculate a T. It's just that T will violate some energy condition. The list you gave, "dust, radiation, or lambda," includes an example that does violate an energy condition. – Ben Crowell Aug 12 '11 at 13:04
Can you provide a reference or link? Thanks. – mtrencseni Aug 12 '11 at 23:22
For example, [WMAP][1]'s conclusions about $\Omega_\Lambda$ support our universe having $\Lambda>0$; other independent sources are also plentiful. See my answer above as to why such a $\Lambda$ violates an energy condition and causes test particles to in repulsive ways. [1]: – Stan Liou Aug 13 '11 at 4:46
Edited the answer to make it clearer. – mtrencseni Aug 13 '11 at 10:06

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