## Advances in Theoretical and Mathematical Physics

- Adv. Theor. Math. Phys.
- Volume 15, Number 4 (2011), 1085-1113.

### Null asymptotics of solutions of the Einstein–Maxwell equations in general relativity and gravitational radiation

Lydia Bieri, PoNing Chen, and Shing-Tung Yau

#### Abstract

We prove that for spacetimes solving the Einstein–Maxwell (EM) equations,
the electromagnetic field contributes at highest order to the nonlinear
memory effect of gravitational waves. In *Nonlinear nature of gravitation and gravitational-wave
experiments,* Christodoulou showed
that gravitational waves have a nonlinear memory. He discussed how this
effect can be measured as a permanent displacement of test masses in a
laser interferometer gravitational-wave detector. Christodoulou derived a
precise formula for this permanent displacement in the Einstein vacuum
(EV) case. We prove in Theorem 2.6 that for the EM equations this permanent
displacement exhibits a term coming from the electromagnetic
field. This term is at the same highest order as the purely gravitational term that governs the EV situation. On the other hand, in Section 3, we
show that to leading order, the presence of the electromagnetic field does
not change the instantaneous displacement of the test masses. Following
the method introduced by Christodoulou in *Nonlinear nature of gravitation and gravitational-wave
experiments,* and asymptotics derived
by Zipser in *The global nonlinear stability of the trivial solution of the
Einstein–Maxwell equations* and *Extensions of the stability theorem of the Minkowski space in
general relativity: Solutions of the Einstein–Maxwell equations*, we investigate gravitational radiation at null infinity
in spacetimes solving the EM equations. We study the Bondi mass loss
formula at null infinity derived in *Extensions of the stability theorem of the Minkowski space in
general relativity: Solutions of the Einstein–Maxwell equations*. We show that the mass loss formula
from *Extensions of the stability theorem of the Minkowski space in
general relativity: Solutions of the Einstein–Maxwell equations* is compatible with the one in Bondi coordinates obtained in *Gravitational waves in general relativity X:
Asymptotic expansions for the Einstein–Maxwell field*.
And we observe that the presence of the electromagnetic field increases
the total energy radiated to infinity up to leading order. Moreover, we
compute the limit of the area radius at null infinity in Theorem 2.7.

#### Article information

**Source**

Adv. Theor. Math. Phys., Volume 15, Number 4 (2011), 1085-1113.

**Dates**

First available in Project Euclid: 11 June 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.atmp/1339438352

**Mathematical Reviews number (MathSciNet)**

MR2929683

**Zentralblatt MATH identifier**

1251.83015

#### Citation

Bieri, Lydia; Chen, PoNing; Yau, Shing-Tung. Null asymptotics of solutions of the Einstein–Maxwell equations in general relativity and gravitational radiation. Adv. Theor. Math. Phys. 15 (2011), no. 4, 1085--1113. https://projecteuclid.org/euclid.atmp/1339438352