# Energy in a volume changing the geometry

Hi All,

As Einstein said, matter causes dirtortion in space. And he also said Energy and matter are equivalent. I would like to know if is there a formally accepted method of knowing how the space would be curved if there were only one photon in space. Supose this photon had energy E associated with its energy by Planck's law, and had a somewhat definite localization, say, a gaussian shape with 1 mm width.

I would also appreciate your indication of some good references on this subject.

Best Regards

DaTario

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pervect
Staff Emeritus
This sounds like a quantum gravity problem to me, not a GR problem. GR deals in continuous distributions of matter and of fields, not point particles. A problem similar to the one that you describe that GR could solve is finding the gravitational field of a microwave cavity (with and without microwaves present) for instance.

One would proceed by finding the stress-energy tensor of the electromagnetic field (and the stress-energy tensor in the cavity!), then solve Einstein's field equations.

The stress energy tensor of the electromagnetic field is described in the following web reference, or several textbooks (just about any full GR textbook should cover this).

http://world.std.com/~sweetser/quaternions/EandM/tensor/tensor.html

One point I'm not quite sure about is how to deal with the stress-energy tensor of the cavity. I think that one would have to include the mechanical stresses, and the surface currents of the cavity, and insure they occupied a finite volume. The mechanical stresses wouldn't be a problem to distribute (assume the cavity is thin walled, and spread them evenly). The surface currents might be more of a problem (one might have to assume a finite conductivity to keep the current density finite).

I'm not really sure how to write the resulting stress-tensor down (the one for the cavity), nor do I recall seeing it discussed in detail in any textbook.

I do know from similar problems that including the stress-energy tensor of the cavity is vital to getting sensible answers.

Staff Emeritus
Gold Member
Dearly Missed
Couldn't you simplify? Take a particular energy photon, assume its momentum is along x, and there is no other source of energy or moentum in the region, and then fill in the momentum energy tensor from that? Expoit symmetries to reduce the order of the problem?

pervect
Staff Emeritus
I guess it would be resonable to simplify to a microwave beam. I think one would need realistic boundary conditions on the beam, ones that satisfied Maxwell's equations, to get good results.

This would involve modelling beam-spread correctly, rather than assuming a perfectly collimated beam. Another way of saying this is that one would need to have the pressure terms in the stress-energy tensor of the beam causing it to diverge realistically, rather than ignoring them.

I still don't think a single photon model would be a good idea.

DaTario said:
As Einstein said, matter causes dirtortion in space.
Actually matter causes a distorsion in spacetime, not simply in space. There's a diffference.

Pete

pmb_phy said:
Actually matter causes a distorsion in spacetime, not simply in space. There's a diffference.

Pete
I agree with you. Thank you.

Regarding what was said before by Mr. Pervect and Mr. SelfAdjoint, all I have to say is thank you, for I really don't have any rigorous contact with GR, to provide any helpful comment. I had the impression that the cavity + field approach may lead to some bizarre results, depending possibly on the shape of the cavity ( also the external part of it ) becoming thus a dificult problem. On the other hand, the field state I proposed may include to much modes and, according to your observations, it turns the problem also into a dificult one. It is not my interest to work on a plane wave. A 3D gaussian pulse would be my favorite photon model for the moment. One problem of its propagating nature is that the distortions produced would be time dependent and essentially non local (in the sense that would follow the pulse path).

I am interested in photons having energies larger or comparable to the energy content of a proton's mass.

I will look up in the web reference Mr. Pervect gave me.

One further question. That book, GRAVITATION, with an apple in the front cover, is the best one in this subject ? Which other would you recommend ?

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ohwilleke
Gold Member
pervect said:
GR deals in continuous distributions of matter and of fields, not point particles.
This is the single most befuddling aspect of GR to me and I've never quite understood the basis for it. Short of modeling the entire universe at once, space looks a heck of a lot more like point particles than it does like continuous distributions of matter. The foreignness of this concept is one of the things that makes the GR equations so very difficult to understand, because concepts like pressure and stress are concepts that, while well defined in situations where mass is continuous (e.g. with solid object or relatively compact fluids), are ill defined from a common sense perspective in a typical galaxy or stellar system which is mostly empty space punctuated with discrete small (relative to the total volume) objects.

Why is this necessary? Could you expand on the concept of a continuous distribution of matter a little?

Staff Emeritus
Gold Member
Dearly Missed
Gravitaion, aka MTW, from the authors' initials, is full of stuff you won't get anywhere else. As an introduction to GR, opinions differ. It didn't help me, and some of the track one/track two stuff in it left me baffled for years. I wish I had encountered differential geometry in another book first. Many people seem to swear by Wald's textbook, but I don't have any experience with it.

I would like to know if is there a formally accepted method of knowing how the space would be curved if there were only one photon in space.
The method is: solve the Einstein Field Equations. The EFE are a nonlinear system of second order partial differential equations in 16 unknowns (each of the 16 equations contains thousands of individual terms).

Crosson said:
The method is: solve the Einstein Field Equations. The EFE are a nonlinear system of second order partial differential equations in 16 unknowns (each of the 16 equations contains thousands of individual terms).
I remember of have read once Einstein saying: "I wish I could have understood better that 16 component tensor..."

It seems to be this tensor he was talking about.

pervect
Staff Emeritus
I'd suggest looking at Baez's

Book list

GR tutorial

introductory paper on GR

Wald and MTW's "Gravitation" are both great books, but neither is probably the best *first* book to read on the topic. See Baez's book list for some other recommendations such as Schutz and D'Inverno. (Unfortunately I haven't read either of them, so I don't have any personal recommendations here). Wald is very concise and mathematical and modern, while MTW is somewhat dated, and enormously sprawling, but MTW has material you just won't find elsewhere.

pervect
Staff Emeritus
ohwilleke said:
This is the single most befuddling aspect of GR to me and I've never quite understood the basis for it. Short of modeling the entire universe at once, space looks a heck of a lot more like point particles than it does like continuous distributions of matter. The foreignness of this concept is one of the things that makes the GR equations so very difficult to understand, because concepts like pressure and stress are concepts that, while well defined in situations where mass is continuous (e.g. with solid object or relatively compact fluids), are ill defined from a common sense perspective in a typical galaxy or stellar system which is mostly empty space punctuated with discrete small (relative to the total volume) objects.

Why is this necessary? Could you expand on the concept of a continuous distribution of matter a little?
If we have a bunch of atoms that are not interacting (except at very close range), it doesn't seem terribly complicated to me to want to replace the details of exactly where every atom is with a sort of average overview, which is a simple scalar, which gives the number of atoms / unit volume.

When all of the atoms are standing still, this is good enough. WHen the atoms are moving, we have to refine our model a bit. Basically, we start to think of fluids, rather than atoms.

Fluids not only have a density rho, they have an average velocity. This average velocity is the average velocity of the atoms in a particular "neighborhood" of the fluid. The density rho is the T_00 component of the stress energy tensor. The average velocity are in the T_0i and the T_i0 components of the stress energy tensor.

(Because the tensor is symmetric, the streamline velocity is contained in both places).

Fluids also have a pressure. A bunch of atoms that stand still and do not interact have no pressure. A bunch of atoms that are all moving in the same direction at the same velocity have a velocity, but no pressure. But suppose we have atoms moving to the left and an equal number moving to the right, so that the average velocity is zero.

We abstract this situation with a pressure - this particular example represents an anisotropic pressure, a pressure that exists in only one direction. Three of these anisotropic pressures (one for each direction) make up the guts of the stress-energy tensor.

The other terrms (the non-diagonal terms) in the stress energy tensor can always be made to vanish, by a proper choice of principle axes. Compare the pressure tensor to the moment of inertia tensor. Both of these are 3x3, and both of them represent an ellipsoid with three principle axes. When you align the axes of your coordinate system to the principle axes of the ellipsoid, you get a simple diagonal tensor with 3 elements. So you can think of the stress part of the stress-energy tensor as describing a "pressure ellipsoid".

pervect
Staff Emeritus
As far as light beams with gaussian profiles go:

Apparently there should be such a thing (see for instance)

http://www.atis.org/tg2k/_gaussian_beam.html [Broken]
http://www.newport.com/servicesupport/Tutorials/default.aspx?id=112 [Broken]

but I don't seem to be able to get $\nabla \cdot E = 0$ for such a profile.

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With the help of Mr. Pervect. I guess I can write a more precise state of the electrical field for which I would like to be able to calculate the space-time distortions effects.

$$E(r,t) = E_0 \exp (-( (r - r_c(t)) / w)^2 ), [\tex] where $r_c (t) = (1, 2, 3) + (ct, 0, 0) [\itex] is the center of this gaussian profile and is moving with velocity c in the x direction. The 3D vector (1,2,3) is to represent the initial position. Now the question can be restated : As the energy in a small volume is a volume integral of E squared, is it possible to calculate space time distortions effects due to this EM pulse with the aid of Einstein equations ? pervect Staff Emeritus Science Advisor DaTario said: With the help of Mr. Pervect. I guess I can write a more precise state of the electrical field There's already trouble, though, and we haven't even gotten to the hard part yet. E has a direction,as well as a magnitude, which must be stated. The direction of E must be perpendicular to the direction of propagation. Finally, [itex]\nabla \cdot E=0$ - assuming a free-space solution (no charges), and nearly flat space-time. Because E must be perpendicular to the direction of propagation, if we assume that if the wave is propagating in the 'z' direction, E must have components only in the x and y direction. [tex] G = (x^2+y^2+(z-c)^2)/w(z,t)^2$$

$$E_x = \exp(-G(x,y,z,t))$$

doesn't work, the divergence of E is not zero (no matter what we assume for w(t))

$$E_x = \frac{x}{\sqrt{x^2+y^2}}\exp(-G(x,y,z,t))$$
$$E_y = \frac{y}{\sqrt{x^2+y^2}}\exp(-G(x,y,z,t))$$

making the E-field have radial symmetry, doesn't work either for the same reason (and note that the field is even discontinuous at x=y=0.

Gaussian beam profiles are apparently supposed to exist, but so far I haven't come across a good enough description of them to write down the electric field.

As the energy in a small volume is a volume integral of E squared, is it possible to calculate space time distortions effects due to this EM pulse with the aid of Einstein equations ?
Long before Einstein, Reimann found that distance on any curved (or not) space could be expressed by a metric. Here are some examples:

$$ds^2=dx^2+dy^2$$

$$ds^2 = dr^2 +r^2 d\theta ^2$$

Hopefully, these two metrics look familiar to you DaTario. They both represent flat space, one is in cartesian coordinates and one is in polar coordinates. It is noteworthy that although the two metrics look different, they both represent the same thing. Here is a metric that expresses distance on the surface of a sphere:

$$ds^2 = R^2d\phi ^2 + R^2 Sin^2(\phi) d\theta ^2$$

The curvature of a this surface (of a sphere) is 1/R^2.

Now lets look at a general metric. x, y and z or theta or whatever, lets call them x1,x2,x3...etc. So in general we have:

$$ds^2 = g_1(x_1,x_2,x_3,x_4)(dx^1)^2+ g_2(x_1,x_2,x_3,x_4)(dx^2)^2 + g_3(x_1,x_2,x_3,x_4)(dx^3)^2...$$

In full generality, we could have terms like dx*dy, instead of them all being dx*dx. If we have four variables dx, dy, dz, and dt we can combine them quadratically in 16 ways (dxdx,dxdy,dxdz,dxdt,dydx,dydy,dydz,dydt...) meaning that there will be 16 metric coefficients g11,g12....etc. These 16 metric components, arranged in a 4x4 matrix, are a representation of the metric tensor.

DaTario, are you aquainted with differential equations? do you know the sollution to y'' + y = 0 ?

When we set up a problem in general relativity where the distribution of energy is known, but the curvature of space is not, we are trying to solve for the metric coefficients. This is analagous to solving for y in the above equation, if you imagine that the above equation has thousands of terms.

Let me say it this way: the math of what you are asking is so hard that no man or machine is capable of finding the solution.

pervect
Staff Emeritus
The strong-field case would indeed be very, very, hard.

It *might* be possible to get some weak-field results, though I'm not sure exactly how involved that will be. So far, I haven't even seen a problem specification that satisfies me, though, which isn't a good sign, as this should be the easiest part of the problem.

One possibiity is to look at some of Mallet's work, except that I don't really trust it - i.e. "Weak gravitational field of the electromagnetic radiation in a ring laser". I think Mallet took too many shortcuts, though.

I think the simplest problem to set up would be one with a lot more symmetry - a spherical "photon gas" in a spherical cavity. If one can treat this as a "photon gas", if one does not have to include any induced electromagnetic effects in the wall of the cavity, but only stress effects, it wouldn't be that hard to do, especially if one restriced it to the weak field case. I'm not sure if one can make this simplification, though. I'm also not sure how interesting this would be to the OP.

pervect said:
Because E must be perpendicular to the direction of propagation, if we assume that if the wave is propagating in the 'z' direction, E must have components only in the x and y direction.

$$G = (x^2+y^2+(z-c)^2)/w(z,t)^2$$

$$E_x = \exp(-G(x,y,z,t))$$

doesn't work, the divergence of E is not zero (no matter what we assume for w(t))
I have the impression that you have forgotten the time on the numerator of G definition.

$$G = (x^2+y^2+(z-ct)^2)/w(z,t)^2$$

Do you agree ?

pervect
Staff Emeritus
I don't see how omitting a time dependence (which I don't think I've done anyway), could change the value of the divergence, which is $$\frac{\partial E_x}{\partial x}+\frac{\partial E_y}{\partial y}+\frac{\partial E_z}{\partial z}$$

If the charge density is zero, the divergence of the electric field should be zero.

Crosson said:
DaTario, are you aquainted with differential equations? do you know the sollution to y'' + y = 0 ?
It seems to be the harmonic osclillator ODE. sin(x) and cos(x) seems to solve it. The general method using exponentials is also familiar.Let me introduce some of my background. It may help the discussion. In my PhD thesis I have worked with montecarlo techniques to solve master equations in QM, which represented a system of couled ODE with someting around 500 equations. Using stochastic methods (quantum jumps) in a 1.5 GHz computer, some simulations require near 1 month of computation time. Is it much more than this we are talking about?

pervect said:
I don't see how omitting a time dependence (which I don't think I've done anyway), could change the value of the divergence, which is $$\frac{\partial E_x}{\partial x}+\frac{\partial E_y}{\partial y}+\frac{\partial E_z}{\partial z}$$

If the charge density is zero, the divergence of the electric field should be zero.
I am not sure you have omitted, I just would like to confirm with you your expression for G in your first note containing it.

I agree that the presence of t may not change your conclusion then. It is just a matter of maintaining an agreement on the used equations.

pervect said:
The strong-field case would indeed be very, very, hard.
Tha's is a bad new as I am interested in strong field situations.

pervect said:
I think the simplest problem to set up would be one with a lot more symmetry - a spherical "photon gas" in a spherical cavity. If one can treat this as a "photon gas", if one does not have to include any induced electromagnetic effects in the wall of the cavity, but only stress effects, it wouldn't be that hard to do, especially if one restriced it to the weak field case. I'm not sure if one can make this simplification, though. I'm also not sure how interesting this would be to the OP.
first : What is OP ?

second: I found your idea very interesting. I am worried, however, with the full characterization of this photon gas. What specfic set of interference will take place in this gas. These interferences, I guess, will be responsible for high field amplitudes during small times in specifc locations inside the gas. This specifc pattern of interferences, generated ultimately by the its spectrum and phase distribuitions, may play an important part in the problem.

Best Wishes

DaTario

pervect
Staff Emeritus
#1 OP = original poster

#2
If you keep the intensities low enough where new particles are not created, non-coherent photons do not interact with each other. Thus if you have a container of photons, you have a container of particles that carry momentum and energy, but don't interact. There should therfore be no "interferences". This suggests that an ideal gas would be a good model for high frequency photons inside a large container (a container much larger than the wavelength of the photons). If the container becomes too small, the idea will probably fail because the photons will unavoidably be coherent (the limiting case is where you have a microwave cavity in it's fundamental mode, in which case you basically have a bose-einstein condensate, where all the photons share the same quantum mechanical state).

I'm not sure how to justify this really rigorously, it should be similar to the way that "geometric optics" is justified.

If the container is large relative to the wavelength, but the electric field becomes strong enough to create particles out of the vacuum, you should still have a "photon gas", it would just be a non-ideal gas.

One also needs for the photons to "bounce off" the container walls without energy loss for the gas model to work - this should also happen with idealized electromagnetic waves inside an idealized conducting container (total reflection).

Let's discuss brieflly on this statement of yours:

pervect said:
Thus if you have a container of photons, you have a container of particles that carry momentum and energy, but don't interact. There should therfore be no "interferences".
I guess interference is not an evidence of interactions between photons. In other words: photons may not interact (in the trajectory sense, i.e., they cannot modify the other's kinematics) but may interefere, i.e., may together create high and low levels of field strengh by mutual superposition. The interference may be experimentally tested. H. Walther at Garching, Germany, Max Planck Institute is working on this subject, searching for the electric profile of a propagating wave through the scattering of slow ions or atoms. He manage to obtain de shape of the wave of nanosecond pulses (with around 10 EM oscillations inside the pulse). Sorry, but I don't have the precise information on this reference.

pervect
Staff Emeritus
As I recall now

http://www.arxiv.org/abs/gr-qc/9909014

is basically where I got the idea from - look at expression 17, for instance, and the brief mention of a photon gas.

I don't think there should be any problems with short wavelengths in a large box with the form of the stress-energy tensor givin by Carlip in his eq 17. There probably would be problems if the box contained a "standing wave".

Regardless, the paper should be interesting in that you don't integrate just the energy density to get the total mass - you integrate the energy density and the pressure. However, the result turns out to be that the pressure terms don't contribute to the total mass.

The reason this happens is that in the weak field case, the positive pressure in the interior of a box containing a hot gas (or photons) is balanced by the negative pressure in the walls, so that the total intergal of the pressure terms is zero. Thus the mass of the box is equal to the total energy divided by c^2, where the total energy is calculated by integrating the energy density over the volume.

If the metric inside the box isn't nearly flat, if g_00 starts to get significantly different than one (significant depends on the experimental accuracy desired), a more complicated formula for mass needs to be used and this simple analysis doesn't work.

See

https://www.physicsforums.com/showpost.php?p=671119&postcount=25

https://www.physicsforums.com/showpost.php?p=293632&postcount=64

for the formula that needs to be used in that case, along with the textbook reference.

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