What causes space curves? Energy or rest mass?

[quasar987]

If it's energy, a photon must curve space.

If it's rest mass a photon doesn't curve space and an object going at speed 0.99c doesn't curve space more than when it's not moving.


A friend of mine asked me this question after asking two of his profs at McGill University and getting two different answers! So.. tensors masters, which is it?

 

[selfAdjoint]

What curves space according to Einstein's field equations is momentum-energy, and its stress. Recall that in relativity momentum and energy are combined into a single four-vector; this four-vector forms the first row and first columns of the 4x4 momentum energy tensor. The other components of the tensor are derivatives of the above wrt the spacetime variables.

 

[Garth]

A photon curves space-time. But because it is trace-free the extra curvature created by adding electromagnetic energy is also trace free so
R = 0 where R is the Ricci or Curvature scalar due to that electromagnetic energy.

Mass and other forms of energy also curve space-time.

So the answer to your question is both.

Garth

 

[DrWarezz]

Hi, sorry to 'barge in on this convo', but could someone explain what is ment by "warped space-time"? \
I'm kinda new to all this, and I've read and heard of it all too many times. I just don't get it.
Anyone? )

Thanks in advance,
[r.D]

 

[Severian596]

At the risk of blowing your mind, we see "warped space-time" as gravity. Yay!

 

[Severian596]

Okay, okay. A diagram is in order at least:

http://www.astronomynotes.com/evolutn/grwarp.gif

This illustrates warped space-time. Space-time is the "sheet" defined by the grid, and the recessed areas in this sheet signify that something of significant mass exists in space-time. The mass warps the surrounding space-time, creating a gravity well. A beam of light moving past a large object is bent according to this well's curvature.

 

[Gonzolo]

A photon curves space-time.

Then does an electric field right? Your answer is much simpler here than your discussion with pervect in the other thread!

Except I don't know what "trace-free" means.

 

[pervect]

Anything that has energy, or that has momentum, curves space-time. This includes electric fields, and electromagnetic radiation.

 

[Antonio Lao]

What causes space curves? Energy or rest mass?

My 2 cents idea is that the cause can be from the product of very strong infinitesimal orthogonal forces applied at a distance comparable to the Planck length. Applying twice give an expression for the square of energy. The square roots of square energy give two solutions as kinetic energy and potential energy. And these energies define a moving mass and a rest mass. Therefore, by simply ignoring the effect of vacuum polarization, curvature is affected by the square of energy derived from rest mass as well as moving mass (momentum).

 

[Garth]

Except I don't know what "trace-free" means.

If you are not familiar with tensors then I wouldn't worry about it.

However if you do...
The (stress-[energy)-momentum] tensor Tuv describes in tensor form the mass, momentum and stress intensities of a system at any point in a field. That tensor for electro-magnetic radiation takes a particular form and when you perform a tensor operation on it, called 'taking the trace', you end up with zero. It is therefore described as being trace-free. This tensor then tells space-time how to curve through another tensor called the Einsteinian Guv, which in turn then tells matter and radiation 'how to move'. There you are GR in a nutshell!

Garth

 

[blue_sky]

Anything that has energy, or that has momentum, curves space-time. This includes electric fields, and electromagnetic radiation.

I'm really interested in your answer. There is experimental evidence of this?

blue

 

[pervect]

I'm really interested in your answer. There is experimental evidence of this?

blue

The best experimental evidence I can think of off the top of my head is the bending of light, with the additional assumption that momentum is conserved. If gravity bends light, there must be a reaction force on the body doing the bending to maintain the conservation of momentum. Also, I believe the advance of the perihelion of mercury's orbitpredicted by GR can be traced to this effect, but I'm not 100% positive about that.

In any event, the assumption that the stress-energy tensor causes the bending of space-time is the heart & soul of Einstein's general theory of relativity. So, if you prefer, add the words "according to General Relativity" to the statement I made.

Einstein's field equations are

\large G_{uv} = 8 \pi T_{uv}

The entity on the left, T_uv, is the stress energy tensor. (The entity on the right is a measure of the curvature of space-time.

The stress energy-tensor is nothing but the density of energy and momentum per unit volume. When you multiply the stress-energy tensor by a vector representing a volume, the result you get is the total energy and momentum in that volume as described by the energy-momentum 4-vector.

(Note, it may not be obvious how to represent a volume by a vector. The way it's done is to take the only vector that's orthogonal to the volume in 4d space-time, this is the "time vector" associated with that volume).

You can also understand the stress-energy tensor as describing the "flow" of energy-momentum, as per Baez's webpage on the stress-energy tensor

http://math.ucr.edu/home/baez/gr/stress.energy.html

 

[pmb_phy]

If it's energy, a photon must curve space.

If it's rest mass a photon doesn't curve space and an object going at speed 0.99c doesn't curve space more than when it's not moving.


A friend of mine asked me this question after asking two of his profs at McGill University and getting two different answers! So.. tensors masters, which is it?

If you're asking "What is the source of gravity?" then the answer is, to quote MTW (page 404) "Mass is the source of gravity." (If this is wrong then someone should send the corrections to the authors explaining the errors of their ways. )

I say this so that people won't think that Wheeler was sloppy or ignorant when he said that "mass tells spacetime how to curve and spacetime tells mass how to move" which is a very famous saying and one that is 100% correct.

But this "mass" is not rest mass. Its "mass-energy" (aka inertial mass aka relativistic mass) i.e. the m in m = E/c². This is like saying that charge is the source of an EM field. Charge in one frame is current stress/pressure in another. Just like current generating a magnetic field, the momentum effects the gravitational force on a particle only when the particle is moving.

And yes - light generates a gravitational field. E.g. http://www.geocities.com/physics_world/grav_light.htm

The only reason people can say that "energy is the source of gravity" is by involking the fact that mass is proportional to energy. This is exactly what Einstein did.

I've yet to see a 'derivation' of Einstein's equation which didn't make this association. Had they assumed that this "mass" in m = E/c² was only rest mass then they'd be making a serious error.

A tensor is the mathematical object which describes mass-energy. Recall that 4-current is the mathematical object that is the source of the EM field. Charge is the time-time component of the 4-current and mass-energy is the time-time component of the energy-momentum tensor.

Note: Mass can generate a gravitational field in the absence of spacetime curvature so its more appropriate to say that mass generates a gravitational field rather than mass generates spacetime curvature.

There will be a strong tendency to think that since the time-time component of the stress-energy-momentum tensor does not have the units of mass density then it can't be mass that generates a gravitational field. But this is hardly a good claim. It'd be similar to claiming that because the time-component of 4-currerent has the units of current density (i.e. same units as spatial component) then charge is not the source of an EM-field.

Pete

ps - Surgery went fine and I'm a whole person again who is relaxing and getting better. I took a cleansing ride/walk downtown to read my e-mail and thought I'd poke my head in.

 

[pervect]

But this "mass" is not rest mass. Its "mass-energy" (aka inertial mass aka relativistic mass) i.e. the m in m = E/c². This is like saying that charge is the source of an EM field. Charge in one frame is current stress/pressure in another. Just like current generating a magnetic field, the momentum effects the gravitational force on a particle only when the particle is moving.

Nitpick time...

It's generally regarded as incorrect to call E/c^2 mass. Pete likes to call E/c^2 "relativistic mass", which I don't have a real problem with, though I would not urge anyone else to copy that practice. Calling E/c^2 mass, though, is wrong (at the very best, it's ambiguous).

As far as I'm aware Pete and I both agree on the physics, that it's the energy E (or if you prefer, the relativistic mass, or to use MTW's phraseology, mass-energy) that curves space-time.

As far as Wheeler goes, I believe that Wheeler's actual quote was not "mass tells space how to curve" but "matter tells space how to curve". But I haven't found a totally definitive source for the quote.

Saying that mass curves space-time is wrong, as "mass" means "invariant mass". To give the usual URL (which Pete has seen once or twice before)

http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html

Does mass change with velocity?

There is sometimes confusion surrounding the subject of mass in relativity. This is because there are two separate uses of the term. Sometimes people say "mass" when they mean "relativistic mass", mr but at other times they say "mass" when they mean "invariant mass", m0. These two meanings are not the same. The invariant mass of a particle is independent of its velocity v, whereas relativistic mass increases with velocity and tends to infinity as the velocity approaches the speed of light c. They can be defined as follows:

mr = E/c^2
m0 = sqrt(E^2/c^4 - p^2/c^2)

where E is energy, p is momentum and c is the speed of light in a vacuum. The velocity dependent relation between the two is

mr = m0 /sqrt(1 - v2/c2)

Of the two, the definition of invariant mass is much preferred over the definition of relativistic mass. These days, when physicists talk about mass in their research, they always mean invariant mass. The symbol m for invariant mass is used without the subscript 0. Although the idea of relativistic mass is not wrong, it often leads to confusion, and is less useful in advanced applications such as quantum field theory and general relativity. Using the word "mass" unqualified to mean relativistic mass is wrong because the word on its own will usually be taken to mean invariant mass. For example, when physicists quote a value for "the mass of the electron" they mean its invariant mass.

 

[blue_sky]

The best experimental evidence I can think of off the top of my head is the bending of light, with the additional assumption that momentum is conserved. If gravity bends light, there must be a reaction force on the body doing the bending to maintain the conservation of momentum. Also, I believe the advance of the perihelion of mercury's orbitpredicted by GR can be traced to this effect, but I'm not 100% positive about that.

In any event, the assumption that the stress-energy tensor causes the bending of space-time is the heart & soul of Einstein's general theory of relativity. So, if you prefer, add the words "according to General Relativity" to the statement I made.

Einstein's field equations are

\large G_{uv} = 8 \pi T_{uv}

The entity on the left, T_uv, is the stress energy tensor. (The entity on the right is a measure of the curvature of space-time.

The stress energy-tensor is nothing but the density of energy and momentum per unit volume. When you multiply the stress-energy tensor by a vector representing a volume, the result you get is the total energy and momentum in that volume as described by the energy-momentum 4-vector.

(Note, it may not be obvious how to represent a volume by a vector. The way it's done is to take the only vector that's orthogonal to the volume in 4d space-time, this is the "time vector" associated with that volume).

You can also understand the stress-energy tensor as describing the "flow" of energy-momentum, as per Baez's webpage on the stress-energy tensor

http://math.ucr.edu/home/baez/gr/stress.energy.html

Pervect, my question was unclear sorry about that. I was asking if there are experimental evidence that also the electric fields, and electromagnetic radiation curve the space-time. I understand that it's true for mass and the exemplum u made are clear. But what about electric fields, and electromagnetic radiation? There are experimental evidence that they curve space-time?

blue

 

[quasar987]

The best experimental evidence I can think of off the top of my head is the bending of light, with the additional assumption that momentum is conserved. If gravity bends light, there must be a reaction force on the body doing the bending to maintain the conservation of momentum.

This is bugging me. Suppose your photon is initially traveling vertically (in the Oxy plane). When it passes the sun, it is deviated horizontally. One could say it gains an horizontal velocity component. But when it gains horizontal component, it must lose some vertical components because (Vx² + Vy²)^½ must be c.

So if you suppose that the photon does't attract the sun in it's turn then it's "partly OK", the total momentum of the photon in conserved (but not his parts). By conserving horizontal momentum, you violate conservation of vertical momentum.

But if you suppose it does attract the sun, the horizontal momentum of the sun-photon system is conserved but the vertical is not.


Something else that is funny: If the photon is coming directly at the sun, say vetically, then the initially at rest sun is attracted towards the photon and it gains a vertical component in its momentum. If you want momentum to be conserved, the vertical momentum of the photon must be altered in the same amount. But the vertical speed of the photon cannot be modified, it is already c. Since p = E/c = hf/c. => the frequency of the photon gets bigger.

Same thing for a photon getting away from the sun, its frequency gets smaller.

 

[pervect]

A photon gains and loses momentum without changing its speed. A photon going into or out of a gravity well changes it frequency. Falling down, it blue-shifts, climbing out of a gravity well, it red-shifts. However, measured by the rods and clocks of a local observer, it will always move at the same speed, 'c'.

 

[quasar987]

Nice.



(I have to enter 10 characters )

 

[pmb_phy]

Nitpick time...

Okay.

It's generally regarded as incorrect to call E/c^2 mass.

That is waaayyyyy wrong. pervect believes that its very rare for somone to define mass in such a way that E = mc² = Kinetic Energy + Rest Energy.

Many people do this in GR. Please note that m is not defined as E/c². In fact Alan Guth does this when he teaches his Early Universe course at MIT. E = m² is an equality between E and m. It is NOT not a definition of m. Mass, m, is defined as p/v. In some circumstances (free/isolated particle/object) it can then be proved that E = mc² for a particle or an isolated object. I.e. it can be proved that E/c² = p/v. If an object has a finite extent and it is not isolated then E/c² does not equal p/v.

Note -

For some reason pervect keeps mixing these up. Why? Do you have a very limited exposure to the relativity literature? To expand your horizons try reading something new, e.g. D'Inverno, Mould, Rindler, etc.

Pete likes to call E/c^2 "relativistic mass", ..

That is inaccurate. I don't like to call p/v "relativistic mass." I prefer to call it either simply "mass" or inertial mass. I use the term "relativistic mass" so people will be less likely to get it confused with "rest mass." I think it is not a good idea to call p/v anything but "mass." But that's another story.

..which I don't have a real problem with, though I would not urge anyone else to copy that practice.

Why would you urge someone to stop thinking along a particular line? It is not logical to assume that since you found it of no use that it is impossible for others to find it useful. There is simply no reason for such urging. Especially since the concept is widely used in the physics literature and is 100% precise and logical when it is used. Taking such a term out of one's language is highly unfruitful and can lead to igorance. If a student does not learn what the m in E = mc² means in all places that he might come into contact with it (i.e. texts, journals etc.) then how do you expect them to understand such texts such as Cosmological Principles, John A. Peacock, Cambridge University Press, (1999) (This is the cosmology text that is used at MIT by the way).

See examples (Especially Peacock and MTW) at http://www.geocities.com/physics_world/relativistic_mass.htm

Calling E/c^2 mass, though, is wrong (at the very best, it's ambiguous).

That is highly incorrect. It is completely inappropriate to call something wrong just because you have personally chosen to use a different definition. It is very clearly not wrong.

..or to use MTW's phraseology, mass-energy) that curves space-time.

Why do you ignore their use of the term "mass" when it doesn't suit you? I've given you examples in the past if I recall correctly. There are clearly places in MTW in which they use the term "mass" to mean E/c². I've shown you those places. If you forgot (or I recalled incorrectly) please see quotes from Peacock and MTW in - http://www.geocities.com/physics_world/relativistic_mass.htm

As far as Wheeler goes, I believe that Wheeler's actual quote was not "mass tells space how to curve" but "matter tells space how to curve".

He has said it in different ways in different places. In fact in his and Taylor's latest book Exploring Black Holes he wrote in the front portion of the book. In - http://www.eftaylor.com/pub/front_matter.pdf the authors write

Spacetime tells mass how to move; mass tells spacetime how to curve ...

His co-author Edwin F. Taylor also quotes that in his acceptance speech for the 1998 Oersted medal presented by the American Association of Physics Teachers, 6 January 1998 - See http://www.eftaylor.com/oersted/

Saying that mass curves space-time is wrong, as "mass" means "invariant mass". To give the usual URL (which Pete has seen once or twice before)

That is an illogical claim since you've chosen a definition for the term "mass" and then claimed that when people don't use the term as you have chosen to use it then they are wrong.

What you've claimed here is inconsistent with the way people use the term in the real world, i.e. what a person means when they write E = mc² (see link to examples above). It is rest mass that is invariant. Why did you neglect to make this clarification?

http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html
http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.htm

That is one personal opinion of the person who wrote that page. Nothing more.

One thing that the page refers to is a quote by Lev Okun where he refers to a comment by Einstein who seems to be saying that E does not equal m². What Einstein actually did is different. Einstein sometimes used "mass" to refer to rest mass and sometimes to refer to what you've called "relativistic mass" which is m = gamma*m₀. One example is where he explains that light has mass (in his book "The Evolution of Physics") and in his book The Meaning of Relativity where he refers to the changing of the inertial mass of a particle when it is in a gravitational field. Lev Okun didn't look in that portion of the book. He only looked in the first portion (I pointed this out to him and he explained to me that he didn't see it when he wrote his articles on mass).

The person who wrote that comment "These days, when physicists talk about mass in their research, they always mean invariant mass." is clearly missleading the reader since advanced cosmology texts don't adhere to that definition. The author is a particle physicist as I recall. He doesn't know how the term is being used in GR and cosmology.

When my back and leg is better and I'm able to travel more I'll see if I can dig some examples up in the GR/SR/Physics journals.

The lack of paying attention to the exact meaning of what mass is can lead to serious errors. Therefore one should understand exactly what they are reading. That is regardless of what term they choose to use in their own life. Griffiths wrote an article on mass renormalization in the American Journal of Physics. His lack of understanding of how mass is defined and the little subtleties that go along with it led him to an invalid conclusion.

People here often claim that mass-energy is not the source of gravity and that mass-energy, stress and momentum is. What they neglect to say is that even the inertial mass of an object, even in SR, is a function of the stress in an object. Even to completely describe a body in SR one must therefore use the energy-momentum tensor and not simply use it only in GR as the source of gravity.

People here also seem to think that one can always replace p/v with E/c² which is clearly wrong in all cases. It is only true under certain circumstances. As an example of when p/v does not equal E/c² - If there is a rod at rest in the inertial frame S and lies on the x-axis and has a rest mass of m₀ then an observer at rest in S' - the inertial frame moving in the +x direction - will not measure the ratio p/v to be identical with E/c². This is one good reason not to confuse relativistic mass (i.e. m = p/v) with energy.

Pete

ps - Please note; I will be unable to respond until next week. I will be using the internet only once a week from a while and even then I might just use it for e-mail. Frankly these same old arguements are getting really boring.

 

[pmb_phy]

If you are not familiar with tensors then I wouldn't worry about it.

However if you do...
The (stress-[energy)-momentum] tensor Tuv describes in tensor form the mass, momentum and stress intensities of a system at any point in a field. That tensor for electro-magnetic radiation takes a particular form and when you perform a tensor operation on it, called 'taking the trace', you end up with zero.

In some cases that is true. But in cases when the radiation is disordered for example the trace is not zero.[/QUOTE]Pete

 

[pmb_phy]

At the risk of blowing your mind, we see "warped space-time" as gravity. Yay!

Actually, curved spacetime is the change in gravity, i.e. spacetime curvature = tidal force.

There is experimental evidence of this?

No. It is a prediction of GR.

But when it gains horizontal component, it must lose some vertical components because (Vx² + Vy²)^½ must be c.

No. That is incorrect. The speed of a photon changes as it moves through a gravitational field. I think that you're confusing this with the fact that the speed of light is the same in all inertial frames of reference. A gravitational field is not an inertial frame of reference. For a simple example please see - http://www.geocities.com/physics_world/uniform_light.htm

Pete

 

[quasar987]

Great website Pete.. I'm bookmarking this.

 

[pervect]

Frankly these same old arguements are getting really boring.

Well, why do you start them then?

I think that GR makes much more sense when it is explained simply by saying that energy curves space-time. This is an oversimplification of course, so I usually explain very shortly therafter that it's actually the stress-energy tensor that curves space-time, not just energy.

Toi say that "mass" curves space-time is ambiguous at best, and in the context of this thread, given the specific question asked, it's wrong.

"What curves space, energy or rest mass". Only one of the two mentioned quanties gives the right answer, and that is energy, not rest mass.

Saying "realtivistic mass" curves space-time would not be wrong, but would be distinctly odd since that's not how the question was asked.

 

[pmb_phy]

Well, why do you start them then?

I don't start them. I simply chime in when someone else does and in this case you were claiming that it was incorrect/bad to think of mass as being relativistic mass. I.e. you constantly say things like "It's generally regarded as incorrect to call E/c^2 mass." when it is simply wrong to do so. In fact I have only a few GR texts whose authors might agree with you. But that greater majoriyt does not agree with you. And that includes MTW, Wald, Peacock, Schutz, Rindler, etc. Each of these in at least one place state in no uncertain terms "mass = energy". I've pointed these clear and obviouys examples out to you and you've ignored them, God only knows why you've ignored these examples and have gone on top claim the opposite. Care to explain that for me?

Some might find it easier to not view it that way or they might find it nice to avoid it. But in no sense of the word can it be claimed that it is "generally regarded as incorrect."

I think that GR makes much more sense when it is explained simply by saying that energy curves space-time.

To you perhaps. But not to all .. or even most.

This is an oversimplification of course, so I usually explain very shortly therafter that it's actually the stress-energy tensor that curves space-time, not just energy.

Let's clear this up - If there is matter at event A then there is curvature at event A. But if there is matter at event A then it does not imply that there is curvature near this event. It may be that at a neabry event there is no curvature. Matter here does not imply curvature over there. Next; One reason that some say that stress contributes to a g-field field is that inertial mass is a function of stress. I.e. if I were to calculate what some people call the "relativistic mass" of a body then that mass, m, would be a function of the stress on the body.

Toi say that "mass" curves space-time is ambiguous at best, and in the context of this thread, given the specific question asked, it's wrong.

It is no wronger to say this than it is to say the charge generates an EM-field since charge in one frame is current in another. In this case mass in one frame is momentum and stress in another. The analogy is most strongest when viewing GR (in the weak field limit) in the paradigm of gravitomagnetism.

Saying "realtivistic mass" curves space-time would not be wrong, but would be distinctly odd since that's not how the question was asked.

[/quote
Its not wrong. It is another way to view/state it. In my opinion its a more accurate way to state it. I.e. 'mass in one frame is stress/momentum in another and as such mass is the source of gravity. The stress-energy-momentum tensor is that mathematical quantity which describes mass in its entirety.

Pete

 

[Kond]

energy loss?

if photons curve space do they lose energy as they proceed?
and if so where would this energy come from?

 

[pervect]

What a blast from the past. 10/12/2004 was the last post, I wonder why this thread suddenly reappeared?

The only way in which a photon, moving through a vacuum space-time, could lose energy would be by the emisssion of gravitational radiation.

I do not believe that this happens, though I'm not aware of any specific references.

 

[JesseM]

My memory is that you also have to take into account "pressure" in determining the curvature of spacetime...can pressure be seen as just a form of kinetic energy or something, or is it really treated fundamentally differently in the equations of GR?

 

[cesiumfrog]

But if you suppose it does attract the sun, the horizontal momentum of the sun-photon system is conserved but the vertical is not.

Something else that is funny: If the photon is coming directly at the sun, say vetically, [...] the frequency of the photon gets bigger.

Regards momentum components, you forgot part of the symmetry (the sun isn't just attracted in one component); there shouldn't be any problems.

And indeed, at the surface of the sun, the stars would look bluer than otherwise. The effect has even been demonstrated in laboratories on earth.

 

[pervect]

My memory is that you also have to take into account "pressure" in determining the curvature of spacetime...can pressure be seen as just a form of kinetic energy or something, or is it really treated fundamentally differently in the equations of GR?

If you have a static system, gravity couples to the energy density + 3*pressure. (This same number comes up with Baez & Bunn's "sphere of coffee grounds" approach, too). The buzzword that applies to the mass of a static system in general relativity is "Komar mass".

This expression leads to an effective "double counting" of the gravitational effects of energy in some situations. For instance, if you had a sphere of relativistically hot gas the gravity at the surface of the shell of the hot gas would be twice that what you'd expect from a Newtonian analysis, due to this double-counting effect. See for instance

http://en.wikipedia.org/wiki/Mass_in_general_relativity

where I write

Imagine that we have a solid pressure vessel enclosing an ideal gas. We heat the gas up with an external source of energy, adding an amount of energy E to the system. Does the mass of our system increase by E/c2? Does the mass of the gas increase by E/c2?

Yes, and no, respectively. Because the pressure vessel generates a static space-time, one can utilize the concept of Komar mass to find its mass, treating the ideal gas as an ideal fluid. Using the formula for the Komar mass of a small system in a nearly Minkowskian space-time, one finds that the mass of the system in geometrized units is equal to E + ∫ 3 P dV, where E is the total energy of the system, and P is the pressure.

The integral ∫ P dV over the entire volume of the system is equal to zero, however. The contribution of the positive pressure in the fluid is exactly canceled out by the contribution of the negative pressure (tension) in the shell. This cancellation is not accidental, it is a consequence of the relativistic virial theorem (Carlip 1999).

If we restrict our region of integration to the fluid itself, however, the integral is not zero and the pressure contributes to the mass. Because the integral of the pressure is positive, we find that the mass of the fluid increases by more than E/c2. Since the fluid is not an isolated system, talking about its mass may be misleading unless great care is taken. This is an example of a non-isolated system with a finite volume. Thus, as explained earlier, the mass of this system is not invariant, and depends on the choice of observational frame. The Komar formula calculates the mass of the gas in its rest frame.

The significance of the pressure terms in the Komar formula can best be understood by a thought experiment. If we assume a spherical pressure vessel, the pressure vessel itself will not contribute to the gravitational acceleration measured by an accelerometer inside the shell. The Komar mass formula tells us that the surface acceleration we measure just inside the pressure vessel, at the outer edge of the hot gas will be equal to G\left(E + 3 P V \right) / r^2 c^2

where E is the total energy (including rest energy) of the hot gas
G is Newton's Gravitational constant
P is the pressure of the hot gas
V is the volume of the pressure vessel.

This surface acceleration will be higher than expected because of the pressure terms. In a fully relativistic gas, (this includes a "box of light" as a special case), the contribution of the pressure term 3 P V will be equal to the energy term E, and the acceleration at the surface will be doubled from the value for a non-relativistic gas.

Note that this Wiki example shouldn't be taken as if it were from an independent source - I'm the original author of it.

Note that the total mass of the isolated system increases by E/c^2. For a closed static system, the pressure terms average out to zero. The increase in the mass of the hot gas given by the pressure terms is exactly equal to the decrease in mass of the pressure vessel given by the tension terms.

 

[pmb_phy]

If it's energy, a photon must curve space.

If it's rest mass a photon doesn't curve space and an object going at speed 0.99c doesn't curve space more than when it's not moving.


A friend of mine asked me this question after asking two of his profs at McGill University and getting two different answers! So.. tensors masters, which is it?

The stress-energy-momentum tensor T is the geometrtical object which dictates the shape of the spacetime. By definition it is active gravitational mass which is the source of gravity. The active gravitational mass density is a function of both energy density and pressure. The details can be found in

http://www.geocities.com/physics_world/mass_paper.pdf

where I explain why this is so. Note: Matter can generate a gravitational field and not yield a curved spacetime. A curved spacetime is the geometrical language of what you heard as tidal forces. You can have a gravitational field without tidal forces. In fact a lot of Einstein's early work on GR he used a uniform g-field, i.e. g-fields in which the spacetime is flat. But then again that is another story accounted for here

http://xxx.lanl.gov/abs/physics/0204044

Best wishes

Pete