An electromagnetic analog
of gravitational wave memory
Midwest Relativity Meeting
Milwaukee, WI Oct. 25, 2013
L. Bieri and DG, Class. Quantum Grav. 30, 195009 (2013)
Odd properties of gravitational waves
• The sensitivity of the detector goes like r-1
• There is a permanent change in the detector (memory) after the wave
has passed
• In addition to the “ordinary” memory which has to do with changes in
the second derivative of the quadrupole moment, there is a
“nonlinear” memory that has to do with the angular distribution of
the energy radiated in gravitational waves.
Is there an electromagnetic analog of
these properties of gravitational waves?
• If a detector of electromagnetic fields simply follows the motion of a
test charge q then its sensitivity goes like r-1
• After the wave passes the charge has undergone a kick
• Weak field slow motion gravity: displacement goes like (projected)
second time derivative of quadrupole moment of source
• Slow motion EM: kick goes like (projected) first time derivative of
electric dipole moment of source
But electromagnetism is a linear theory
so there can’t be an analog of “nonlinear”
memory. Or can there?
In GR in addition to the contribution from energy radiated in gravitational
waves, there is exactly the same contribution from energy radiated in
electromagnetic waves and in neutrinos.
So maybe the “nonlinear” memory is really a “null” memory and has to do
with all sources of energy that can get to null infinity.
To find the electromagnetic analog, we have to consider charge that can get
to null infinity (as happens e.g. in the case of a massless charged scalar field).
Expand Maxwell’s equations in powers of
EA=XA + …
Er=W r-2 + …
Ju=-L r-2 + …
SA= ∫ XA d u
Dv=qSA/(mr)
-1
r
How does the kick depend
on the charge radiated?
SA=S1A+S2A
S1A depends on W(∞)-W(-∞)
(ordinary kick)
S2A depends on ∫ L d u
(null kick)
Let alm be the spherical harmonic expansion coefficients of ∫ L du
Then we have
S2A = ∑l>0 (-4 p/l(l+1)) alm DA Ylm
The same relation holds between S1A and W(∞)-W(-∞)
Conclusions
• There is an electromagnetic analog of gravitational wave memory; but
in this case the memory is a kick rather than a distortion
• Even though Maxwell’s equations are linear, there is an analog of the
“nonlinear” memory for gravitational waves
• This is because what matters in this case is not the nonlinearity but
the fact that the memory is due to sources that can get to null infinity
• Thus the “nonlinear” memory is really a “null” memory