1. An electromagnetic analog of gravitational wave memory Midwest Relativity Meeting Milwaukee, WI Oct. 25, 2013

2. L. Bieri and DG, Class. Quantum Grav. 30, 195009 (2013)

3. 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.

4. 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

5. 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).

6. Expand Maxwell’s equations in powers of r -1 E A =X A + … E r =W r -2 + … J u =-L r -2 + … S A = ∫ X A d u  v=qS A /(mr)

7. How does the kick depend on the charge radiated? S A =S 1A +S 2A S 1A depends on W(∞)-W(-∞) (ordinary kick) S 2A depends on ∫ L d u (null kick)

8. Let a l m be the spherical harmonic expansion coefficients of ∫ L du Then we have S 2A = ∑ l >0 (-4  / l ( l +1)) a l m D A Y l m The same relation holds between S 1A and W(∞)-W(-∞)

9. 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