How Does General Relativity Connect Mass and Geometry?

[aeroboyo]

From my very limited knowledge of General Relativity, i have the impression that mass is just considered as 'other' in this theory. By that i mean, GR assumes matter to be made of something other than space-time, and only deals with the effect that matter has on the geometry of space-time. Is that a correct interpretation of GR? Does GR offer any explanation of what matter is at a fundamental level?

Maybe this question doesn't warrent an entire thread. I'm curious about the connection between mass and space-time in GR. I started wondering about this after reading about a theory called Heim Theory which tries to explain matter at the most fundamental level as being essentially due to the geometry of space-time, i.e like some kind of 'twising' of space-time. Think of a tornado... it appears to be an object with a structure but it's really just made out of air like its surroundings. If the geometry of space-time can sucessfully explain gravitation, then why not use geometry of space-time to explain other forces, or even matter on a fundamental level? Why does GR restrict itself to gravitation and assume mass to be 'other'...

 

[dextercioby]

Nope. GR uses classical "matter fields" and describe their gravitational interaction. Matter fields are "input" (more specifically their energy momentum tensor) for the Einstein field equations.

Daniel.

 

[aeroboyo]

but what is a classical matter field?

 

[dextercioby]

Any space-time function which transforms under a finite-dimensional linear representation of the full Lorentz group when its argument (x^{\mu}) undergoes a Lorentz transformation and its Lagrangian density doesn't exhibit gauge symmetry.

Daniel.

 

[aeroboyo]

...but of course...

 

[pervect]

From my very limited knowledge of General Relativity, i have the impression that mass is just considered as 'other' in this theory. By that i mean, GR assumes matter to be made of something other than space-time, and only deals with the effect that matter has on the geometry of space-time. Is that a correct interpretation of GR? Does GR offer any explanation of what matter is at a fundamental level?

Maybe this question doesn't warrent an entire thread. I'm curious about the connection between mass and space-time in GR. I started wondering about this after reading about a theory called Heim Theory which tries to explain matter at the most fundamental level as being essentially due to the geometry of space-time, i.e like some kind of 'twising' of space-time. Think of a tornado... it appears to be an object with a structure but it's really just made out of air like its surroundings. If the geometry of space-time can sucessfully explain gravitation, then why not use geometry of space-time to explain other forces, or even matter on a fundamental level? Why does GR restrict itself to gravitation and assume mass to be 'other'...

The simplest explanation is that it is not mass that causes gravity in GR. Rather, it is energy and momentum (in the form of the stress-energy tensor, as other posters have indicated).

The idea that it is mass that causes gravity is basically a not-so-useful carryover from Newtonian gravity. In Newtonian gravity it is mass that causes gravity. In GR, any form of energy will interact gravitationally.

Mass in GR is actually somewhat tricky - one definition of mass doesn't quite cover all situations, so GR has several different defintions. This sort of complexity does not afflict the stress-energy tensor, which has only one definition, which can be described as the density of energy and momentum per unit volume. Note that it takes several numbers, 16 numbers of which 10 are unique, to describe the stress-energy tensor at a given point in space-time, not just one.

 

[aeroboyo]

thanks pervect for the clarification... i sort of get it now. Is there any fundamental reason in nature why it takes 16 numbers (of which 10 are unique) to describe the stress-energy tensor, or is it just a mathematical convenience/neccesity. It seems to me that if there are several definitions of 'mass' but only one of the stress-energy tensor, then the 'density of energy and momentum per unit volume' must be more fundamental to nature than the concept of mass.

 

[dextercioby]

The symmetry of the stress-energy tensor is not required in classical field theory in flat space-time (i.e. in the absence of gravity), but usually matter couples to gravity by a term metric-stress-energy tensor and thus the only the symmetric part of the tensor is involved.

However, possible couplings of gravity to matter in the context of general relativity is a tricky subject.

Daniel.

 

[aeroboyo]

I'm going to read 'A First Course In GR' before i post any more threads on the subject... otherwise i'd probably just be asking too many elementary questions. But, if anyone wants to clarify on one point, then feel free...

Einstein relates the stress-energy tensor to curvature right... doesn't that mean they are in essence the same thing. That energy and momentum are bent space-time. If that is a correct interpretation then my logic tells me that GR preety much says that matter is bent space-time also, since energy is related to matter by e=mc^2. Just my, probably, misguided 2 cents.

 

[jgraber]

I'm going to read 'A First Course In GR' before i post any more threads on the subject... otherwise i'd probably just be asking too many elementary questions. But, if anyone wants to clarify on one point, then feel free...

Einstein relates the stress-energy tensor to curvature right... doesn't that mean they are in essence the same thing. That energy and momentum are bent space-time. If that is a correct interpretation then my logic tells me that GR preety much says that matter is bent space-time also, since energy is related to matter by e=mc^2. Just my, probably, misguided 2 cents.

John Wheeler seriously considered the idea that mass-energy is "nothing but" the curved spacetime under the name of "geometrodynamiics". Technically, one way to say this is since G mu nu is always equal to T mu nu, then G mu nu is identical to T mu nu.

Wheeler eventually gave up on this idea because there are many different forms of T mu nu, ie electromagnetic, strong, weak, even rest mass versus various forms of energy, that have different properties and are hard to deal with using just four dimensional spacetime. However, if you introduce a fifth dimension, you can deal with electromagnetism geometrically. This is called Kaluza-Klein theory.

So your idea is not completely crazy, but the simpler forms of it do not work.
Hope this helps.

Jim Graber

 

[yogi]

The notion of mass defined by particulate chunks having physical dimensions would seem to be inconsistent with all that is known about subatomic interactions. If there were a fundamental particle - what would it look like? - how could it be described? - the notion of a distortion of space-time seems to be the most promising contender if one is restricted to a 3 space one time dimension universe. John Wheeler commented on several occasions that he still believed that all matter could be made of electrons and positrons. If these can be modeled as spacio-temporal vorticies as many have suggested, then matter is at some level no different than Space-time. I am not familiar with Hiem Theory, but it sounds like it is worth reading.

 

[aeroboyo]

Interesting that it seems to be not possible to 'geometricize' mass-energy in only R^4. Heim apparently managed to geometricize all known types of particles, but that required R^6. Those two extra dimensions are not spatial or rolled up into a tiny ball like in Kaluza-Klein theory however, they are just associated with organisational properties... i.e they are 'informational', not spatial or temporal. That's a novel concept i think, that there is another type of dimension other than spatial and temporal.

I think Heim theory would be well worth reading. Unfortunately i can't because my math isn't good enough (he basically invented his own math because in his theory he never deals with infintismals) his books are written in German and i wouldn't want to study a 'far out' theory like that until i have a solid grounding in more established theories like GR. Here is a brief abstract for anyone interested about Heim's idea that matter might be purely made up of space-time itself. The 'metron' referred to is the fundamental, smallest area that can exist in nature according to Heim. Hence his theory never deals with infintismals.

"The essence of Heim’s theory is its complete geometrization of physics. By this is meant the fact that the
universe is pictured as consisting of innumerable small, locally confined geometric deformations of an otherwise
unpertubed 6-dimensional metronic lattice. The influence these deformations have on our 4-dimensional world,
or the effects of their projections into it, constitute the structures we interpret as gravitons and photons, as well as
charged and uncharged particles. The theory ultimately results in a formula from which the masses of all known
elementary particles and a few unknown ones may be derived. In addition, it provides a picture of cosmology
differing widely from the established one. Despite the insight gained into particle physics, the theory is not entirely equvalent to modern quantum theory. For this reason Heim has extended the theory to 12 dimensions. Only this extension allows full quantization, and as a consequence it becomes possible to unite relativity and quantum theory. Even 6 dimensions are not
sufficient to accomplish this."

Different particles are made by the deformation of different combinations of dimensions of the 'metron'. Here's a brief summary:

a) The first type is a lattice deformation involving only the 5th and the 6th coordinates. In the 4 remaining
dimensions the metronic lattice remains undisturbed. Physically, this may be interpreted as a structure
existing in the two transdimensions. Since our senses are not attuned to events in the two
transdimensions this may difficult to visualize. Although the deformation exists in dimensions 5 and 6 only, and does not project directly into our 3 dimensions, its effect may be occasionally be felt in the rest of the world. Under certain conditions it
may be extended into the four remaining dimensions in the form of quantized gravitational waves, so called
gravitons.

b) The second type of deformation again involves dimensions 5 and 6, and in addition time, the 4th
dimension. Again, this particle like structure does not project directly in our 3-dimensional world, but
is felt here only in the form of waves. Heim derives the property of these waves and shows that they
are identical to those of electromagnetic light waves or photons. It follows that case (b) describes a
particle like structure in the 4th, 5th and 6th dimensions, extending into the remaining 3 dimensions in
the form of photons.

c) The third possible deformation involves 5 dimensions, i.e. all coordinates except time. This 5-
dimensional structure projects into 3-dimensional space of our experience, i.e. it forms a condensation
here, and it is reasonable to assume that we are sensitive to such condensations. This is indeed the case,
and Heim shows that they give rise to uncharged particles with gravitational mass and inertia.

d) The final deformation involves all 6 coordinates. This again leads to condensations in the space of our
experience, giving rise to particles, but as in case (b), the inclusion of time leads to electric phenomena
as well. Heim can show that 6-dimensional lattice distortions lead to charged particles.

I guess this highlights what jgraber said, about Wheeler having found it difficult to explain the many different types of mass and energy using only R^4. If you add extra dimensions it might be possible though. However since GR exists in R^4 i guess any attempt to geometricize other properties of the world other than gravity would have to require extra dimensions... which is one thing that string theory, LQG, and Heim theory all have in common i think.

 

[yogi]

"The great theories of physics have geometric origins" Ed Witten

 

[masudr]

Heh. What does Ed Witten know?!

 

[aeroboyo]

well GR is a great theory and it's all about geometry... Were SR and GR the first theories in physics to model a force in a geometric way? I can't think of any other examples, obviously there are many that came after GR like Kaulza-Klein, Strings, LQG etc. Seems like around about 1910 was the novel idea to explain a force of nature in terms of geometry.

 

[Daverz]

Ever looked at Newton's Principia? Tons of geometry.

 

[aeroboyo]

Einstein obviously managed to relate space-time content with space-time curvature sucesfully... I wonder if he was thinking about describing the other forces and perhaps even matter as space-time curvature, maybe that is the unified field theory he was searching for. If gravity can be described as space-time curvature then i'd like to see a logical argument which says that the other forces should or could not be.

Although this is where philosophy steps in i think. If mathematics requires extra dimensions to describe the world in a geometric way (as in some kind of bent space-time) then should we therefore conclude that the world has to have these extra dimensions. Maybe geometry and mathematics requires these extra dimensions but maybe our tools of description don't reflect reality, if you know what i mean. I think some people think of adding dimensions as a 'fudge factor' and they hold that against a theory. I'm aware of the Dutch theoretical physicist G. 't Hooft's holographic principle which says that all the information contained in some region of space can be represented as a`Hologram' - a theory which `lives' on the boundary of that region. So maybe, for example, someone could 'reduce' Kaluza-Kelin's theory from R^5 into an R^4 theory using some principles of holography... the other tennent of the principle is that in a given volume, there is an upper limit to the density of information about the whereabouts of all the particles which compose matter in that volume, which suggests that matter itself cannot be subdivided infinitely many times; rather there must be an ultimate level of fundamental particles. That strikes me as being exactly what Heim proposed, that there is a fundamental minimum AREA in space-time, which he termed the Metron. Interesting that both his theory and this principle require a minimum area... my cranky thought for the day!

Anyway, I'm looking foreward to the day when I'm competent enough to read through theories like LQG for that reason. I think geometry is a beautiful way to describe the universe. I haven't read through Newton's Principia...i'm sure it has a lot of geometry but when i say geometry i really mean bent space-time, in this context.

 

[yogi]

Einstein's success with explaining gravity as curvature did lead to many attempts in his later years to find a similar relationship between geometry and charge - but as we know he was not successful in this endeavor - in fact, the success of GR may have biased his approach - in the case of charge, the conditioning of space may be some sort of dynamic as proposed by Maxwell and later by Dirac, rather than a static distortion.

 

[aeroboyo]

Very interesting... i guess space-time curvature might very well be dynamic at the quantum level and only appear to be static at the macroscopic level. Hence a theory like GR which assumes space-time is static would model the macroscopic level or reality very well but not the quantum level.

I don't know anything about Maxwell or Dirac's theories. It is interesting that according to Heim the only difference between charged particles, and uncharged particles with gravitational inertia, is the exclusion of a distortion of the temporal dimension in the latter case. Of course the distortion is dynamic in Heim's theory... if the distortion is a repeating pattern then the particle is permanent or if it doesn't repeat then that particle is virtual and only exists a short time. Anyway i appolgise for talking about Heim's ideas so much in a GR forum but i think they are maybe compatible, like two sides of the same coin.

PS you can bet that to desrcibe a dynamic distortion of space-time would involve some crazily complex math!

 

[aeroboyo]

yay...

my boas, wheeler, frankel, geroch and schutz all arrived today... so i can start reading ;) The spacetime physics is indeed the origonal red edition, with answers at the back etc. Probably won't be posting anything here for a while as i read through those, unless I'm really stuck. I'd like to thank everyone who helped me decide which books to get, i think these are the right ones to begin with.