What would GRT look like if negative masses existed? What would be the "Schwarzschild" metric, etc.?
This question amounts to asking "if the laws of physics didn't apply, what would the laws of physics say about <fill in anything you like>?"
Search the forums for "negative mass" for many previous threads on various aspects of this idea.
And as a follow-on to Jonathan's suggestion, there is always a list at the bottom of each thread of 5 threads that have similar titles and I've usually found those to be a good place to start.
Yes, I had searched through some of those. But my interest is in why we don’t use the exquisite math in GRT to describe “spacetime(s)” in electrodynamics, where there is repulsion in addition to attraction? So I wondered if anyone knew how “gravity” would look if masses repelled each other instead of attracting.
You're asking two different questions here.
We don't use spacetime curvature and its associated formalism to describe electromagnetism because that approach only works for forces whose strength is exactly proportional to the mass the force is acting on (this is one way of thinking about the equivalence principle). Electromagnetism is not such a force; two objects of the same mass can have different charges and hence experience different forces and accelerations. Gravity will never you give different accelerations for different objects at the same place.
As Phinds has said above, there is no way of using science to answer this question. It's like asking which animal would be the natural prey of elephants if elephants were carnivorous - it's an interesting question, but you should ask it of a science fiction writer not a biologist.
We can pursue the first question here, but the second is out of scope for PhysicsForums.
First, mass doesn't enter into GR directly. It enters in only as the norm of the energy-momentum four-vector, and so the original question boils down to "how would GR look if lengths were negative?" This can't be answered, as lengths aren't negative.
The next question, of using GR math for E&M, is answerable two ways. One is that the GR formalism is a) overkill and b) inappropriate - inappropriate because E&M is a vector theory and GR is a tensor theory. The other is that there is a section in MTW where they cast E&M in a way where it fits better into GR, but it's still rather unwieldly.
To phrase this question within general relativity, one would probably look at the energy conditions. http://en.wikipedia.org/wiki/Energy_condition
Violations of the most common energy conditions are discussed in http://arxiv.org/abs/gr-qc/0001099.
Negative energy and negative mass are two different things. The latter implies (as stated by the OP) a repulsive 'gravitational charge' , which is not at all the same as negative energy or energy condition violations. The latter lead to hypothetical scenarios where spacelike transport is possible (but not repulsive gravity).
Well, I don't know if negative mass is defined in GR, since mass itself is not really defined, so I was just suggesting a way to make it well-defined. But if one wants to go more in the spirit of "repulsive gravity", how about a cosmological constant for the accelerating expansion, or exotic matter for wormholes?
Exotic matter is what I was referring to. It violates energy conditions, may lead to worm holes or several other ways to have effectively spacelike transport, and can be considered negative energy but not mass. I'm not aware that it leads to repulsive gravity.
Cosmological constant is interesting but constant - not something you could get a lump of.
I've never seen any speculative proposal within the math of GR that has a two bodies responding oppositely to the influence of a third (without bringing in other charges). Exotic matter might respond differently but not in opposite sense compared to normal matter.
How about proposals where the cosmological constant comes from some form of matter, say like quintessence?
If you have any reference to such application, it would be interesting. Given the ocean of papers published on speculative approaches to FTL using exotic matter, I would think that if someone had conceived of a speculative formulation where quintessence could lead to an anti-gravity device (for example), they would have published a paper about it.
Plug exotic matter into Raychaudhuri's equation.
I don’t understand why you guys (some of you at least) brush the question aside. Here’s how a serious answer might help a self-taught beginner like me, and maybe others, understand the theory better. I’ve worked my way all the way through the math but don’t understand the big picture very well yet – obviously. I don’t know exactly where I told the math that Galileo’s two objects fell DOWN rather than UP. Working backwards, it seems that the signs of the Christoffel symbols would have to at least flip in the geodesic equations so that, for example, a photon grazing the sun would curve away from it rather than toward it. Also it seems that the potential energy would flip sign. It would approach infinity as the radius goes to zero, and approach zero at r goes to infinity. So change (-r_s/r) to (+r_s/r) I suppose? When I get a block of time, I’ll study that and see what happens. So I’ll work it out myself, thanks anyway.
PS. Yes I know that trying to analyze electrodynamics with this math would be complicated, maybe even impossible. But again it would facilitate understanding for me.
I think the closest answer is exotic matter discussed in the last several posts. If you express the stress energy tensor (the source term for gravity in GR) in a local orthogonal frame at some event, the 00 term is normally positive and corresponds to total energy density. You can, within the math, make it negative instead. This is exotic matter. The implication of George Jones comment is that in addition to FTL effects I'm familiar with, you can get some repulsive gravity behavior as well.
However, in contrast to EM, there is no analog of charge (a conserved scalar quantity) that is a source of gravity, so there is no way within the math to simply reverse gravitational attraction by charge reversal. Mass itself is not not fundamental in GR, it cannot even be defined in the general case (including, in realistic cosmologies). This is the sense behind the response: you can't express this idea in GR.
To get repulsion you would have to flip the sign of the gradient of the gravitational time dilation factor, which is the square root of first term in the metric:
But I'm not sure what that implies for the spatial metric components under the given boundary conditions, and if there can be a valid metric at all like this.
By Birkhoff, it would be impossible for spherical, non-rotating, body to produce such a metric, since the solution is unique (for cosmological constant zero). With a cosmological constant, you could get repulsive effects, but they would not appear to originate with the central body.
I know Bondi pubished some papers on the topic of negative mass in GR, for instance
Negative Mass in General Relativity Rev. Mod. Phys. 29, 423 – Published 1 July 1957
I think I've seen Bondi's paper at one time, but it doesn't seem to be publically available anymore. I believe that considering the topic eventually led him to create the so-called Bondi mass.
Google finds some other remarks by Bonnor as well. General Relativity and Gravitation, Volume 21, Issue 11 , pp 1143-1157
There appears to be some concern about run-away motion if positive and negative masses coexist from what I can gather from what fragments of the papers are on the WWW. I believe that you get very bad behavior if you allow a gas consisting of negative mass particles to interact with a gas made of positive mass particles, though I don't recall the source of the argument. The argument ias I recall it s that that thermodynamically, the negative temperature negative-mass-particle gas would gain negative energy and the positive-mass-particle gas would gain positive energy. Thermo isn't my best subject, but it does seem to me to maximize the total number of states - the positive mass gas has positive kinetic energy and a positive temperature, the negative-mass gas has negative kinetic energy and a negative temperature, and the negative temperature is "hotter" than the positive temperature. (See for instance the wiki article on negative temperature). Thus you get the run-away effect where the entropy is maximized by a high positive energy and positive temperature for the positive mass gas, contributing a postive number of states / entropy S = Q/T, and you get an equal contribution from the negative Q and the negative T from the negative mass gas.
Anyway, I believe there is a fair amount in the literature on negative mass, but it's all rather old. and my recollection is that the main concerns are rampant instability if you have negative mass interacting with positive mass.
Do you recall how they define negative mass? Is it just negative 00 component of SET (which most authors would call exotic matter), or do they derive altered field equations (thus more like alternate GR rather than negative mass in GR)? The one page of Bonnor that I can see without pay from your link does not specify, but it seems the assumptions in the abstract would lead to different field equations (thus not what I would call GR).
Separate names with a comma.