Anti-gravitation -- fun to think about! Here's a problem I see with it:
All stuff in GR with no other forces acting on it travels along geodesics. Even postulated anti-gravitating stuff is going to have to travel along geodesics in GR -- what else could it do? You could try saying it's going to run backwards on a geodesic, but that's still a geodesic. So the conclusion I come to is all stuff has to travel along geodesics, and behave like normal gravitating matter. Unless there are other forces at work. Which means no anti-gravity, at least in GR.
you have asked exactly the right question! Here is the answer: in a curved space, anti-gravitating matter does not move on geodesics. Geodesics are defined through the covariant derivative. The derivative depends on the transformation property of the quantity to be differentiated.
When you drop the assumption that the gravitational energy (momentum) has to be identical to the kinetic energy (momentum), it turns out that there are two quantities which you could want to preserve under parallel transport.
The one gives the usual geodesics, the other a different - but well defined - curve (in flat space, both agree). This becomes possible because the anti-gravitating fields do not transform as 'usual' tensor fields under general diffeomorphism. Nevertheless, their transformation behaviour is known from the representation it belongs to, and it can be used to define an appropriate covariant derivative.
This is not so different from gauge-theories. E.g. in an electromagnetic field, a positron moves on a different curve than an electron does. It has a different coupling to the gauge-field, which shows up in the covariant derivative.
In the Newtonian limit, you can interpret the covariant derivative of the particle as the conservation of total energy. A usual particle falls down on earth, thereby it gains kinetic energy and lowers its gravitational energy. An anti-gravitating particle would do the same when falling 'up'. Or, with sufficient initial momentum directed towards earth, it would fall down, but thereby loose kinetic energy and gain gravitational energy.
Best,
B.
2:51 PM, April 20, 2006
Garrett said...
Ahh, so you have to give up the nice action for a free particle -- it has to have an action other than the proper time of the path. The new free particle action must explicitly include the connection, just like for a charged particle interacting with other gauge forces. And you have to give up the equivalence principle.
Inertial mass no longer must equal gravitational mass...
Hey, you know what? This stuff sounds exactly like the teleparallel formulation of gravity. That's effectively equivalent to GR, but allows for particle equations of motion with unequal gravitational and inertial mass. Are you familiar with that?
you are absolutely right, I have to give up the action one commonly uses to derive the geodesic equation as the curve of a particle.
However, also in usual GR, to get the curve of the particle one does not need to postulate an extra-action for the particle. Though not widely done in textbooks, it is possible to derive the curve directly from Einsteins's field equations, using the energy momentum tensor of a pointlike particle. If you use this approach you find two possibilities for the curve, depending on the structure of the energy momentum tensor - it is either that of the usual particle, or that of the the anti-gravitational particle. This corresponds to either parallel transporting the 'usual' or the 'new' kinetic momentum.
I stumbled across teleparallel gravity several times, but never looked really into it. Thanks for the remark, I will have a closer look.
Best,
B.
2:01 PM, April 21, 2006
garrett said...
It makes sense that you could get the motion from the energy-momentum tensor. And you could probably get the particle action by contracting that tensor with g. (I like actions, can you tell?)
I'd probably get a better idea of what you're doing if I had your paper available, which I don't. :( I'll have access to the UCSD library in a couple of months though, and I'll look at it then.
The basic idea of teleparallel gravity is to treat gravity as a force field in flat Minkowski space. So, there's a metric (or rather a vierbein) and a connection, but its curvature is zero. The torsion of the connection, though, is not zero, and the dynamics are in there. It's a pretty theory, and agrees with GR predictions. But I (and most) still like good 'ol geometric GR -- but I'm always open to whatever works best in the big picture.
Here's an intro to teleparallel GR that includes the equation of motion for free particles with different inertial and gravitational mass: http://arxiv.org/abs/gr-qc/0410042
Probably worth a quick look from you just to see if you spot similarities.
I had a brief look at the paper about teleparralel (TP) gravity. I can't say I really understand it, but it seems to me there is no relation to my model. The only similairity is that in the TP it is possible to have an inertial mass other than the graviational. However, the price to pay for this seems to me quite high.
In the TP, the ratio between gravitational and inertial mass is a continuos parameter, connected to a continuous deformation of the Geodesic motion. (Not sure whether the deformed curves still have a geometrical meaning).
In my model, the ratio is discrete, it is either +1 or -1, it is more a charge symmetry than a deformation. The crucial point, which does not appear in the TP, is the modified transformation behavior of the new fields. The cause for the non-standard-geodesic motion in the TP is a completely different one.
BTW, I read yesterday the Physics Today from April and found there is a letter (from R.E.Becker) with an comment on TP, as well as the corresponding answer from S. Weinberg, in case you are interested.
Best,
B.
9:19 PM, April 24, 2006
Garrett said...
Hi Sabine, I have no great love for teleparallel GR. (I'm flexible about whether torsion exists or not, but I don't think it should exist at the expense of curvature!) I just thought it might be relevant, since it's compatible with having two different connections and corresponding different mass "charges." But I guess not.
I'm going to go post a question about your paper on PF, where I can write math that doesn't make me barf.
2:24 AM, April 25, 2006
Today, I gave a telephone interview to a very nice young man for the print version of Welt der Wunder about black holes - apparently a topic of fatal attraction. To my amazement, it turned out, he was born the same day as I, in the city where my parents still live (and no, Lubos, I am not a schizophrenic hermaphrodite - at least not that I know).Puzzled by the guy knowing my birthday, I googled my name - and I was surprised to find (besides my birthday in the google-cache) this article about my recent work on Anti-Gravitation Assumptions are repulsive, especially about gravity Tom Siegfried which gives a very accurate brief summary of the idea:"Electric charges can be positive or negative. Opposite electric charges attract each other; like charges repel. Gravity, on the other hand, seems to be always attractive -- all masses move toward each other. Apparently all matter has a positive gravitational charge, and like gravitational charges attract. If there was such a thing as negative mass, it would behave in the opposite way and push away all ordinary matter, a prospect that most physicists find repulsive -- except for Sabine Hossenfelder. "So, for those of you who always want to know more, here is the story about anti-gravitation that is not in the paper: Briefly after I finished high school and started studying at Frankfurt University, I moved to the Bockenheim, where the old campus was located. Unfortunately, I made a wrong move and ended up lying in bed the next two weeks. Whenever I turned my head, several nerves between my head and toes went up in sudden fireworks. Essentially, I stared at the ceiling for two weeks, wishing I had some anti-gravitational device to lift my furniture. Besides shouting at my boyfriend and eating too much chocolate, I started wondering why there are no negative "charges" in general relativity. At this time, I was blissfully unaware of several reasons why this would not be possible, the most pressing one being the stability of the vacuum, others you find addressed in the Anti-Gravitation FAQs.I am afraid, I annoyed my roommate with my anti-gravitation theories for quite a while. And I was stubbornly insisting I did not understand his concerns why this would not make sense. (Anti-gravitation was not the only 'theory' I kept discussing with him. At some point we named them SH's Incredible Theories - SHIT. ) Then I got distracted from the anti-gravitation for the next years. Well - I had to write a diploma thesis, and make my PhD. But the topic kept coming back whenever I had some free time. Just that the time was never enough to make it publishable. My last summer was very frustrating in many regards, I was absolutely certain my research would never go anywhere, and I would be unemployed by next year. While I was conference hopping in Europe, I wrote together what conclusions I had so far about the anti-gravitation. And I put it on the arxiv, praying it would be enough, hoping that someone might find it sufficiently interesting to finish what I wasn't able to finish. The reactions on the paper were: none. Though the first version of the paper was crowded with typos, it turned out to be a very useful basis to annoy other people by handing them a copy, and asking for objections. There were far less objections than I was prepared to discuss. And those that came were easily to answer. The most common reaction was: "uhm. interesting. uhm. why don't you talk to. uhm... [someone else]." The most reasonable discussion I had for a long time was with one of my ex-boyfriends (credits go to Andi). Who is not even a physicist, but who understood far more of the idea, and the problems, than several over-educated brains before him. Meanwhile, the guy who I shared the apartment with when I had my back problems, and who I had bugged several years with my anti-gravitational obsession, got a postdoc position at Perimeter Institute. During a phone conversation last October, anti-gravitation slipped in (again), and after listening very politely to my lengthy explanations, he said "uhm. interesting. uhm. why don't you talk to. uhm, Lee." I thought, great idea, much better some guy at PI thinks I am nuts than the people I have to talk to every day. So I sent my paper to Lee. I am not sure whether he actually read it. But the conversations I had with him, and several others, since that time where encouraging enough for me to answer several depressing referee reports, and to send the paper to PLB, after it was rejected by PRD.The revised version which was accepted for publication last month, is considerably improved, and I am enormously grateful for all the discussions I had with the two Stefans, Jim, Lee, Petr and (in the early days...) with Marcus. Unfortunately, the version in PLB is extremely shortened. I have been told repeatedly that the content is indeed much clearer now, after several revisions and shortenings, than in the first version. Though that might be true, imo the latest version suffers from a severe lack of applications. Needless to say, there is work to be done.I am still not completely convinced whether there is some point I have been missing all the time, but I guess I will never find out unless I keep on discussing the idea. The ansatz has several problems that I know of -- probably I am the one most familiar with all of them -- and every second day I get stuck in a dead end. The other days though, everything makes perfect sense. I am still waiting for objections.
"Anti-Gravitation"
8 Comments -
Anti-gravitation -- fun to think about! Here's a problem I see with it:
All stuff in GR with no other forces acting on it travels along geodesics. Even postulated anti-gravitating stuff is going to have to travel along geodesics in GR -- what else could it do? You could try saying it's going to run backwards on a geodesic, but that's still a geodesic. So the conclusion I come to is all stuff has to travel along geodesics, and behave like normal gravitating matter. Unless there are other forces at work. Which means no anti-gravity, at least in GR.
4:23 AM, April 20, 2006
Dear garrett,
you have asked exactly the right question! Here is the answer: in a curved space, anti-gravitating matter does not move on geodesics. Geodesics are defined through the covariant derivative. The derivative depends on the transformation property of the quantity to be differentiated.
When you drop the assumption that the gravitational energy (momentum) has to be identical to the kinetic energy (momentum), it turns out that there are two quantities which you could want to preserve under parallel transport.
The one gives the usual geodesics, the other a different - but well defined - curve (in flat space, both agree). This becomes possible because the anti-gravitating fields do not transform as 'usual' tensor fields under general diffeomorphism. Nevertheless, their transformation behaviour is known from the representation it belongs to, and it can be used to define an appropriate covariant derivative.
This is not so different from gauge-theories. E.g. in an electromagnetic field, a positron moves on a different curve than an electron does. It has a different coupling to the gauge-field, which shows up in the covariant derivative.
In the Newtonian limit, you can interpret the covariant derivative of the particle as the conservation of total energy. A usual particle falls down on earth, thereby it gains kinetic energy and lowers its gravitational energy. An anti-gravitating particle would do the same when falling 'up'. Or, with sufficient initial momentum directed towards earth, it would fall down, but thereby loose kinetic energy and gain gravitational energy.
Best,
B.
2:51 PM, April 20, 2006
Ahh, so you have to give up the nice action for a free particle -- it has to have an action other than the proper time of the path. The new free particle action must explicitly include the connection, just like for a charged particle interacting with other gauge forces. And you have to give up the equivalence principle.
Inertial mass no longer must equal gravitational mass...
Hey, you know what? This stuff sounds exactly like the teleparallel formulation of gravity. That's effectively equivalent to GR, but allows for particle equations of motion with unequal gravitational and inertial mass. Are you familiar with that?
1:48 PM, April 21, 2006
Hi garrett,
you are absolutely right, I have to give up the action one commonly uses to derive the geodesic equation as the curve of a particle.
However, also in usual GR, to get the curve of the particle one does not need to postulate an extra-action for the particle. Though not widely done in textbooks, it is possible to derive the curve directly from Einsteins's field equations, using the energy momentum tensor of a pointlike particle. If you use this approach you find two possibilities for the curve, depending on the structure of the energy momentum tensor - it is either that of the usual particle, or that of the the anti-gravitational particle. This corresponds to either parallel transporting the 'usual' or the 'new' kinetic momentum.
I stumbled across teleparallel gravity several times, but never looked really into it. Thanks for the remark, I will have a closer look.
Best,
B.
2:01 PM, April 21, 2006
It makes sense that you could get the motion from the energy-momentum tensor. And you could probably get the particle action by contracting that tensor with g. (I like actions, can you tell?)
I'd probably get a better idea of what you're doing if I had your paper available, which I don't. :( I'll have access to the UCSD library in a couple of months though, and I'll look at it then.
The basic idea of teleparallel gravity is to treat gravity as a force field in flat Minkowski space. So, there's a metric (or rather a vierbein) and a connection, but its curvature is zero. The torsion of the connection, though, is not zero, and the dynamics are in there. It's a pretty theory, and agrees with GR predictions. But I (and most) still like good 'ol geometric GR -- but I'm always open to whatever works best in the big picture.
Here's an intro to teleparallel GR that includes the equation of motion for free particles with different inertial and gravitational mass:
http://arxiv.org/abs/gr-qc/0410042
Probably worth a quick look from you just to see if you spot similarities.
3:07 PM, April 21, 2006
SHIT happens!
interesting idea, keep us updated.
11:13 PM, April 21, 2006
Hi garrett,
I had a brief look at the paper about teleparralel (TP) gravity. I can't say I really understand it, but it seems to me there is no relation to my model. The only similairity is that in the TP it is possible to have an inertial mass other than the graviational. However, the price to pay for this seems to me quite high.
In the TP, the ratio between gravitational and inertial mass is a continuos parameter, connected to a continuous deformation of the Geodesic motion. (Not sure whether the deformed curves still have a geometrical meaning).
In my model, the ratio is discrete, it is either +1 or -1, it is more a charge symmetry than a deformation. The crucial point, which does not appear in the TP, is the modified transformation behavior of the new fields. The cause for the non-standard-geodesic motion in the TP is a completely different one.
BTW, I read yesterday the Physics Today from April and found there is a letter (from R.E.Becker) with an comment on TP, as well as the corresponding answer from S. Weinberg, in case you are interested.
Best,
B.
9:19 PM, April 24, 2006
Hi Sabine,
I have no great love for teleparallel GR. (I'm flexible about whether torsion exists or not, but I don't think it should exist at the expense of curvature!) I just thought it might be relevant, since it's compatible with having two different connections and corresponding different mass "charges." But I guess not.
I'm going to go post a question about your paper on PF, where I can write math that doesn't make me barf.
2:24 AM, April 25, 2006