# Stress-energy tensor

I'm curious what is the interpretation of spatial components of stress-energy
tensor in case of Klein-Gordon field or electromagnetic field. Time-time component
is the energy density of the field, time-spatial components are the momentum density.
The diagonal spatial components are probably preassures in different spatial direction
(but what is the preassure of field???).

Thanks for help.

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pervect
Staff Emeritus
The classical pressure in the electromagnetic field is sometimes called the Maxwell stress tensor, IIRC. E&M fields do have pressure, the magnets that blew up at Cern did so because of the stresses caused by the electromagnetic fields

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The classical pressure in the electromagnetic field is sometimes called the Maxwell stress tensor, IIRC. E&M fields do have pressure, the magnets that blew up at Cern did so because of the stresses caused by the electromagnetic fields
I have reservations in assigning stress energy values to these fields, even though it's demonstratively possible to construct a stress-energy tensor out of electric and magnetic fields. I'm sure no one has actually measured the deviation in trajectory of a charge-neutral particle in the vicinity of strong electromagnetic fields. And with the energies involved, nothing would be certain even if it was measured. The interpretations would be endless within the domain of quantum field theory.

Edit: The previous was, obscure, wasn't it?

How will we say the the electromagnetic stress-energy tensor, as the result of a single electron of an atom dropping down an orbital, to be absorbed at some future time, is located? We have one answer if we consider that the entire wave spreads out in all directions.

After the event, where we measure the electromagnetic energy has arrived at another particular atom, do we have a different result?

Or perhaps the whole idea of force carriers effecting spacetime curvature is wrong, and in the case of electromagnetism, it is only the vector potential acting on charge density, A/\J that posses stress energy.

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Dale
Mentor
Hi Phrak, I think your objections really require a quantum theory of gravitation to answer. There is an effective field theory version that should work at the energies you have described but I don't really know enough about it to comment further.

However, the OP didn't mention anything about photons or any quantum effect, so I think a purely classical answer is sufficient for the OP, and classically it is clear that the Maxwell stress tensor can generate very destructive stresses.

bcrowell
Staff Emeritus
Gold Member
I'm sure no one has actually measured the deviation in trajectory of a charge-neutral particle in the vicinity of strong electromagnetic fields.
Actually this has essentially been done, although not quite as directly as you're imagining.

FAQ: Does light produce gravitational fields?

The short answer is yes. General relativity predicts this, and experiments confirm it, albeit in a somewhat more indirect manner than one could have hoped for.

Theory first. GR says that gravitational fields are described by curvature of spacetime, and that this curvature is caused by the stress-energy tensor. The stress-energy tensor is a 4x4 matrix whose 16 entries measure the density of mass-energy, the pressure, the flux of mass-energy, and the shear stress. In any frame of reference, an electromagnetic field has a nonvanishing mass-energy density and pressure, so it is predicted to act as a source of gravitational fields.

There are some common sources of confusion about this. (1) Light has a vanishing rest mass, so it might seem that it would not create gravitational fields. But the stress-energy tensor has a component that measures mass-energy density, not mass density. (2) One can come up with all kinds of goofy results by taking E=mc^2 and saying that a light wave with energy E should make the same gravitational field as a lump of mass E/c^2. Although this kind of approach sometimes suffices to produce order-of-magnitude estimates, it will not give correct results in general, because the source of gravitational fields in GR is not a scalar mass-energy density, it's the whole stress-energy tensor.

Experimentally, there are a couple of different ways that I know of in which this has been tested. An order of magnitude estimate based on E=mc^2 tells us that the gravitational fields made by an electromagnetic field is going to be extremely weak unless the EM field is extremely intense.

One place to look for extremely intense EM fields is inside atomic nuclei. Nuclei get a small but nonnegligible fraction of their rest mass from the static electric fields of the protons. According to GR, the pressure and energy density of these E fields should act as a source of gravitational fields. If it didn't, then nuclei with different atomic numbers and atomic masses would not all create gravitational fields in proportion to their rest masses, and this would cause violations of Newton's third law by gravitational forces. Experiments involving Cavendish balances[Kreuzer 1968] and lunar laser ranging[Bartlett 1986] find no such violations, establishing that static electric fields do act as sources of gravitational fields, and that the strength of these fields is as predicted by GR, to extremely high precision. The interpretation of these experiments as a test of GR is discussed in section 3.7.3 of [Will 2006]; in terms of the PPN formalism, if E fields did not act as gravitational sources as predicted by GR, we would have nonzero values of the PPN zeta parameters, which measure nonconservation of momentum.

Another place to look for extremely intense EM fields is in the early universe. Simple scaling arguments show that as the universe expands, nonrelativistic matter becomes a more and more important source of gravitational fields compared to highly relativistic sources such as the cosmic microwave background. Early enough in time, light should therefore have been the dominant source of gravity. Calculations of nuclear reactions in the early, radiation-dominated universe predict certain abundances of hydrogen, helium, and deuterium. In particular, the relative abundance of helium and deuterium is a sensitive test of the relationships among a, a', and a'', where a is the scale-factor of the universe. The observed abundances confirm these relationships to a precision of about 5 percent.[Steigman 2007]

Kreuzer, Phys. Rev. 169 (1968) 1007

Bartlett and van Buren, Phys. Rev. Lett. 57 (1986) 21

Will, "The Confrontation between General Relativity and Experiment," http://relativity.livingreviews.org/Articles/lrr-2006-3/ [Broken], 2006

Steigman, Ann. Rev. Nucl. Part. Sci. 57 (2007) 463

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bcrowell, where in the above link is discussion of gravitation from nuclear electric field densities?

I'm skeptical that the reference in question on nuclear gravitational energy is capable of distinguishing the energy-momentum of charge-current in a 4-vector electromagnetic potential from the energy-momentum represented in the the electromagnetic stress energy tensor.

Second, the electromagnetic stress energy tensor contains terms of the Poynting vector. The Poynting vector is the temporal component of *(G/\*dG). G = *F, F = dA. It looks innocent enough, but on closer examination, it expands to terms containing nonzero charge and current density. It's not an equation about the vacuum condition at all, but energy density requiring the presence of charge and/or current!

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bcrowell
Staff Emeritus
Gold Member
bcrowell, where in the above link is discussion of gravitation from nuclear electric field densities?
It's Will's interpretation of the Kreuzer experiment. The link is made in more detail in this 1976 paper by Will: http://articles.adsabs.harvard.edu//full/1976ApJ...204..224W/0000224.000.html

I'm skeptical that the reference in question on nuclear gravitational energy is capable of distinguishing the energy-momentum of charge-current in a 4-vector electromagnetic potential from the energy-momentum represented in the the electromagnetic stress energy tensor.

Second, the electromagnetic stress energy tensor contains terms of the Poynting vector. The Poynting vector is the temporal component of *(G/\*dG). G = *F, F = dA. It looks innocent enough, but on closer examination, it expands to terms containing nonzero charge and current density. It's not an equation about the vacuum condition at all, but energy density requiring the presence of charge and/or current!
The 1976 paper by Will has a lot more of the technical details. I would be interested in seeing what you think after looking it over. It basically uses a toy model of the nucleus, but he argues that the unrealistic modeling of nuclear physics doesn't affect his conclusions.

I read some, as best I could, of your latest linked paper. Very impressive in scope and detail. To fully appreciate it, I would have to go back a few layers of supporting papers. It may be interesting to pursue.

My focus has been on the electromagnetic contribution to the contravariant stress-energy tensor of equation 19, page 227,

$$T^{ab} = * - \frac{1}{4\pi} \left( F^{a}\:_{c} F^{bc} - \frac{1}{4}g^{ab} F_{cd} F^{cd} \right)$$

where the electromagnetic contributions to spacetime curvature are located in the electric and magnetic fields in exclusion of charge and current.

[Come to think of it, as soon as the first derivative of $T^{ab}$ is taken equation (18), this may no longer be true, as charge and current densities might be inadvertently introduced to the vacuum.]

Alternatively, we might consider that the electromagnetic contribution is located in the electrodynamic momentum.

Carver A. Mead, Collective Electrodynamics, introduces this term in a semiclassical treatment of quantum mechanics where the momentum of a particle has an additional contribution from both its charge and the electric field to which it is subject; equation (1.25)

$$\vec{p}_{total}= \vec{p}_{el} + m \vec{v}= q_0 \vec{A} + m \vec{v} \ .$$

Sommerfeld (1952), Electrodynamics has this generalized in a Lorentz invariant quantity

$$S = J \cdot A = \left( \vec{J} \cdot \vec{A} - \rho V \right)\ ,$$

he calls the Schwarzschild invariant, as told by Mead.

This is provocative, but it’s just one term, and it isn’t gr covariant. However, other combinations of the one-forms *J and A are available such as *J/\A that are gr covariant without connections and seem better candidates to me as elements to construct the stress energy tensor of electromagnetism. I would be surprised if a tensor made of these would yield a quantitatively different result from that of Will where he locates the relevant energy in the static electric field--if it could be done.

I've only seen it written as established fact; do you know how the electromagnetic stress energy tensor is motivated or derived?

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Shear Tensor

I Want formula to find shear tensor in general relativity