Re:Does the Electromagnetic field have a Gravitational field?

tessel_at_tum.bot
Date: 01/08/05 

Date: Sat, 8 Jan 2005 22:04:14 +0000 (UTC)

To immediately answer the question in the subject line: of course it does!

Rob Woodside <rwmw@telus.net> wrote:

> Since the electromagnetic field carries energy and momentum with
> Maxwell's stress-energy tensor, one thinks of it as a real thing capable
> of gravitating. Currently Einstein Maxwell theory,
                  ^^^^^^^^^
Is that a pun?

(The term "Einstein-Maxwell solutions" is sometimes used to refer to
Lorentzian spacetimes equipped with suitable tensor fields describing a
charge/current distribution and an EM field, such that a curved spacetime
version of the Maxwell equations are satisfied, and such that the EFE is
also satisfied, where the only term contributing to the
stress-momentum-energy tensor comes from the energy-momentum of the EM
field itself. But as FrediFizzx <fredifizzx@hotmail.com> guessed, the
gravitational effects of man-made fields are unfortunately far too small
to measure in the lab, and even astrophysical EM fields generally
gravitate too weakly to noticeably affect the gravitational field,
although there are interesting proposed exceptions to this rule, like
cosmological magnetic fields.)

But seriously, if you meant to imply that what gtr says about how Weyl
curvature couples to Ricci curvature is somehow subject to interpretation
(or even subject to change), that certainly isn't true. Gtr might well be
wrong about what it says here, but there is no doubt what it does say.

> like the rest of general relativity, puts no constraints on the Weyl
> conformal piece of curvature that is the gravitational field.

Careful! (I discussed this very point in a recent post in which I offered
an overview of the EFE, and we have discussed the same point on numerous
past occasions in this group...)

I think you are thinking of the fact that the Riemann tensor is the sum of
its "traceless part", the Weyl tensor, and its "traced part", a fourth
rank tensor built from the Ricci tensor (or equivalently, from the
Einstein curvature tensor). And of course the EFE directly characterizes
the latter, without explicitly mentioning the former. This is all true---
but it doesn't mean that the EFE "puts no constraints on the Weyl tensor"!
If that were true, long range gravitational interactions would be
impossible, because the immediate presence of matter here and now (which
according to the EFE creates Ricci curvature here and now) could not curve
up a surrounding vacuum region, because the Weyl curvature would not
"couple" to Ricci curvature. Similarly, suitably wiggling matter here
could not create gravitational radiation capable of propagating through a
vacuum.

In fact, the EFE plus the differential Bianchi identity implies a
differential equation which -does- couple Weyl curvature to Ricci
curvature. This "propagation equation" is discussed in various places,
including the monograph by Hawking and Ellis and section 4.4 of the
excellent new textbook:

author = {Sean Carroll},
title = {Spacetime and geometry: an introduction to general relativity},
publisher = {Addison-Wesley},
year = 2004}

The equation I am referring to is (4.90) in that book, which I highly
recommend to anyone who is unfamiliar with the relationship between Weyl
and Ricci curvature in gtr.

> Why doesn't the electromagnetic field carry a piece of the conformal
> tensor explicit in the electromagnetic field just as it carries
> Maxwell's stress-energy tensor which is explicit in the electromagnetic
> field?

Not sure I understood the question, but roughly speaking, if you suddenly
vary a distribution of charges, which is concentrated in some compact
region, in such a way that the distribution of EM field energy varies
aspherically and rapidly, then you can expect to create outgoing
gravitational radiation, which will accompany the outgoing EM radiation.
(The fact that the gravitational wave "signal" and EM wave "signal" move
in lockstep, at least in a electrovacuum, is the best way to understand
what we mean by saying that in gtr, "EM and gravitational radiation
propagate at the same speed".)

To avoid any possible misunderstanding, I'll add a caveat: in gtr, under
suitable circumstances it -is- possible to create an EM wave which is not
accompanied by any gravitational radiation: the gravitational effect of
such an EM wave is "pure Ricci". In such a wave, the immediate presence
of EM field energy-momentum arriving at each event results in Ricci
curvature there and then, but without any accompanying Weyl curvature.
This follows, essentially, from the fact that the strongest EM radiation
is -dipole- radiation, but the strongest gravitational radiation is
-quadrupole- radiation.

Here is a simple explicit exact solution exhibiting the propagation--- not
the generation--- of such a wave. It is called the "SG17 EM wave" (type
17 in the classification of Sippel/Goenner, with a suitable EM vector
potential thrown into the mix). In a "harmonic chart", it can be written:

         ds^2 = -phi du^2 - 2 du dv + dx^2 + dy^2,

        phi = m (x^2 + y^2)/u^2,

        A = sqrt(m) log(u) @/@x

        0 < u < infty, -infty < v,x,y < infty

Here, phi defines the gravitational field, and A (the EM vector potential)
defines the EM field.

To study this solution, it is convenient to define a frame field and to
compute various scalars and tensorial quantities wrt this frame. This
corresponds to choosing a family of ideal observers (one at each event)
and studying their physical experience in detail. Equivalently, one could
choose an NP tetrad and compute with the Newman-Penrose formalism, but the
idea of a frame field is probably easier to grasp. For example, one
possible frame field is

        e_1 = 1/sqrt(2) (@/@v + @/@u - phi/2 @/@v) (timelike)

        e_2 = 1/sqrt(2) (@/@v - @/@u + phi/2 @/@v) }
                                                     }
        e_3 = @/@x } (spacelike)
                                                     }
        e_4 = @/@y }

where these vector fields have unit squared length and are mutually
orthogonal (hence an alternative term, ONB for "orthonormal basis"), and
where not all structure constants vanish (hence another alternative term,
"anholonomic basis").

(In the past, we've studied the interpretation of this particular frame,
in terms of the appearance of the light cones in our chart. We've also
studied this solution using another chart and an inertial frame--- the
frame above is not inertial, so it's rather artificial, but it serves to
make the point.)

This gives the following EM field (as measured by our ideal observers):

        E = sqrt(m/2)/u e_3 B = -sqrt(m/2)/u e_4

This "null" EM field gives rise to a stress-momentum-energy tensor (as
measured by our observers):

                    m [ 1 1 0 0 ]
        T^(ab) = -------- [ 1 1 0 0 ]
                 8 pi u^2 [ 0 0 0 0 ]
                          [ 0 0 0 0 ]

Of course, this "immediate presence" of EM field energy-momentum should
give rise to Ricci curvature at each event. And indeed, the EFE is
satisfied, so we have an "exact Einstein-Maxwell solution", or more
precisely, a "null electrovacuum". However, the Weyl tensor -vanishes-, so
this is a -conformally flat- spacetime--- there is no accompanying
gravitational radiation.

In the past, we've extensively discussed pp waves in this group. We've
contrasted the tidal acceleration of initially static test particles when
"hit" with a suddenly arriving SG17 EM wave versus say an EK4
gravitational wave (this is the one which in a suitable limit gives the
Aichelburg/Sexl "ultraboost" of a Schwarzschild object, which we have also
frequently discussed here in great detail--- see for example the current
thread titled "Mass increase and relativistically moving black holes").
We have described interesting optical effects summarized by the slogan,
"you can't see the wave train enter the station, but you can see it
depart"--- namely, by looking at the distortion of the -optical
appearance- of objects between yourself and the departing wavefronts!
(Exercise: what kind of distortion would you expect for the SG17 example
above? Bear in mind that the Weyl tensor of this spacetime vanishes, but
be careful!) We've discussed the importance of the observation that -all-
the polynomial scalar curvature invariants vanish identically for pp
waves, which is analogous to the fact that in Maxwell's theory of EM, both
polynomial invariants of F_(ab) can vanish without having the field
vanish--- this is just what we mean by saying that we have a "nonzero but
null" EM field above. And so forth.

You can look for those past threads (search under keywords); for the SG
classification of nonvacuum pp waves by their Killing vector fields, see

author = {R. Sippel and H. Goenner},
title = {Symmetry Classes of pp-waves},
journal = {Gen. Rel. Grav.},
volume = 18,
year = 1986,
number = 12,
pages = {1229--1243}}

where you will also find a citation to an earlier paper by Ehlers and
Kundt classifying vacuum waves by their symmetries. (In the threads in
this group, some minor oversights in these papers was corrected.)

whopkins@csd.uwm.edu said:

> The electromagnetic field is scale-invariant, as are Maxwell's
> equations. So it can't provide any non-zero contribution to the Weyl
> tensor.

drl <antimatter33@yahoo.com> commented:

> This is only half right - EM is conformally invariant when there is no
> matter present. As Einstein showed in reference to Weyl's theory, an
> electron in a conformally invariant 4-world would have a mass that
> depended on its spacetime history.

Actually, you're -both- wrong!

-Maxwell-'s equations are conformally invariant--- but not the
Einstein-Maxwell equations! As we have seen, in gtr, Ricci curvature
-does- couple to Weyl curvature, and almost anything you do to generate EM
waves is likely to also produce (much, much, muuuuch weaker) accompanying
gravitational waves.

Drl: look back at the SG17 null electrovacuum above, and observe that you
can easily "add" comoving gravitational radiation (nonlinear
superposition, of course) without adding "matter". This will introduce
nonzero Weyl curvature. You can look for past posts here discussing for
example the Baldwin/Jeffery plane waves (search for that keyword) to see
the general nonlinear superposition of EM and gravitational radiation in
pp waves.

As for Weyl's attempt to unify EM and gravitation, which is now widely
cited as the first explicit example of a gauge theory, this was indeed
immediately shot down by Einstein using the argument you stated, but this
theory is -not- the same thing as Einstein-Maxwell theory, which is simply
Maxwell's theory of EM with the gravitational effects of the EM field
taken into account in the manner demanded by the EFE.

highborn@gmail.com commented:

> It will be a long while before

[curvature due to the energy of an EM field]

> is directly detected. However, some important matters of principle would
> be sorted out if the electromagnetic field did carry a unique
> gravitational field.

Why do you believe the EM field does not induce a "unique" (well-defined?)
gravitational field? We -are- all discussing gtr here, correct? Are you
perchance thinking of factor ordering problems? If so, can you be more
explicit?

"T. Essel"

(spelunking somewhere in cyberspace)

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