# Graviton mass and memory

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## Abstract

Gravitational memory, a residual change, arises after a finite gravitational wave pulse interacts with free masses. We calculate the memory effect in massive gravity as a function of the graviton mass \((m_g)\) and show that it is discretely different from the result of general relativity: the memory is reduced not just via the usual expected Yukawa decay but by a numerical factor which survives even in the massless limit. For the strongest existing bounds on the graviton mass, the memory is essentially wiped out for the sources located at distances above 10 Mpc. On the other hand, for the weaker bounds found in the LIGO observations, the memory is reduced to zero for distances above 0.1 Pc. Hence, we suggest that careful observations of the gravitational wave memory effect can rule out the graviton mass or significantly bound it. We also show that adding higher curvature terms reduces the memory effect.

## 1 Introduction

There is a very natural question that one can ask about the gravitational waves that have been detected by the LIGO/VIRGO: did the detectors leave a permanent effect on the waves or the waves left the detectors intact as they entered? Of course, one can formulate the problem in just the opposite way: did the waves leave a permanent effect on the detectors? The second formulation is better because we might be able to measure the effect if that is the case. It turns out, for certain gravitational waves, part of the strain can be considered as a sort of permanent effect on the detector. This phenomenon is aptly called the *gravitational memory effect* and comes in two related forms: ordinary (or linear) [1], null (or nonlinear) [2] which could be measured soon in the observations.

*g*which needs no coordinates to be defined. General Relativity (GR) is intrinsically four dimensional and the full metric of

*spacetime*manifold

*g*does not really evolve in time: it is what it is. So, if we had known how to obtain all the local observables from the metric for all physically relevant situations, we would not need any further nomenclatures such as memory effect, gravitomagnetism

*etc.*But, since as local observers, we do not have full access to the fully consistent spacetime, it pays to see spacetime as space evolving in time, namely, to see spacetime as a history of space. Such a dynamical picture requires a choice of time and other coordinates and leads to interesting phenomena and the gravitational memory is one such an event: the wave that enters the interaction with the detector masses differs in some well-defined sense from the wave that leaves the interaction. The best way to see the difference is to measure the change in the relative separation of the masses as this is related to the change in the wave profile. In geometric units, in GR, the total change of the wave profile is given by two parts

*r*is the radial coordinate of the detector located at a distance far away from the sources. \(\Delta \) before the summation denotes the difference after and before the wave interacts with the detector, the TT index refers to the transverse-transpose component while the indices

*a*and

*b*are abstract spacetime indices [4]. \(\theta _i\) is the angle between the velocity \(v_i\) and \({\hat{r}}\). The second part in (1) is somewhat more subtle and was initially found by Christodoulou [2] as a result of carefully studying the change at the null infinity once a null stress energy-tensor reaches the null infinity. A more transparent physical interpretation was given by Thorne [5]: considering each graviton emitted by the source as an unbound system, one should simply modify (2) to take into account the gravitons as

In this work, we calculate the gravitational memory as a function of the graviton mass and suggest that a possible observation of memory can constrain or possibly rule out the graviton mass. We shall also discuss the memory effect in quadratic gravity. A priori one would expect that the effect of having a massive graviton (with mass \(m_g\)) amounts to a change of the physically relevant quantities such as the \(\frac{1}{r}\) potential to \(\frac{e^{-m_gr}}{r}\), which indeed is true but the overall factor is not correct: the weak field limit of the Newtonian potential in the massive gravity is \(V(r)=-\frac{4G}{3}\frac{e^{-m_gr}}{r}\), which has the well-known Van-Dam–Veltmann–Zakharov (vDVZ) [6, 7] discontinuity that cannot be remedied by redefining the Newton’s constant as that would lead to a wrong prediction of light deflection by the Sun. Relativistic counterparts of the vDVZ discontinuity have been found recently [8, 9] where massive gravity predicts a maximized total spin for two interacting bodies, while Einstein’s gravity predicts a minimum total spin. Here we study the effects of graviton mass and quadratic terms on gravitational memory and show that it is significantly different from that of GR in the case of massive gravity.

The lay out of the paper is as follows: In section II, we calculate the memory effect in the low energy massive gravity theory (namely the Fierz–Pauli theory). The computation boils down to solving the geodesic deviation equation in the presence of the Riemann tensor which is determined by a passing gravitational wave in massive gravity. In section III, we carry out a similar calculation for quadratic gravity, that has a massive spin-0 and a massive spin-2 particle along with the Einsteinian massless spin-2 particle. The computation is in generic *D* dimensions. In the Appendix, we consider the massive scalar field case to set the notation and our conventions, especially how we define the sources that create the fields.

## 2 Memory effect in massive gravity

*r*, we can take \(m_g r \rightarrow 0\) and the Yukawa decay part reduces to the usual Einsteinian 1 /

*r*form, but the noted discrete difference survives and an accurate measurement of memory can distinguish massive gravity from GR as \(\Delta _{ a b}(m_g \rightarrow 0) \ne \Delta _{a b}(\text {GR})\). On the other hand, if \(r = 1\) Mpc, then one has \(m_g r \approx 1.55\) and the memory is reduced by 0.21 due to the Yukawa decay part. For larger separations, as in the case of the first black hole merger observation which was at a distance \(440^{+160}_{-180}\) Mpc [14], all the memory is wiped out in massive gravity. For weaker bounds on the graviton mass, such as the one noted in [15] (\( m_g < 7.7 \times 10^{-23}\) eV), the memory is wiped out virtually above 0.1 Pc !

## 3 Higher derivative gravity

*D*dimensions:

*r*and \({\bar{\beta }}_{ab}\) is exactly like \({\bar{\alpha }}_{ab}\), except one replaces \(\text {``out''}\) with \(\text {``in''}\). By using (17), the finite relative change in the displacement between two free test particles can be computed as

## 4 Conclusions

Recently gravitational memory effect received a renewed interest [11, 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] for various reasons some of which are: its related to black hole soft hair, asymptotic symmetries and its potential observation in the gravitational wave detectors. Here, we calculated the gravitational memory as a function of graviton mass and showed that for the graviton mass \(m_g \le 10^{-29}\) eV, the memory is significantly reduced for distances beyond 1 Mpc as in the first observation of two black hole mergers which was at a distance of more than 200 Mpc. Moreover massive gravity leaves a discretely different memory on our detectors from the expected general relativity result. The result is summarized by Eq. (20). In the LIGO/VIRGO observations of gravitational waves, memory effect is already in the data but it is hard to distinguish it is from the background noise. In the near future, one might expect to see this effect observed (possibly in eLISA). This observation might rule out massive gravity. We have also calculated the memory effect in quadratic gravity and showed that due to the massive spin-2 mode, the memory is reduced from that of the Einstein’s theory. Here we have used the linearized massive gravity theory which is valid in the weak-field regime that is relevant for the gravitational wave bursts observed on earth. Of course one can consider non-linear extensions of massive gravity such as the one given in [26] but, the above result is universal in the weak field limit as the non-linear extensions reduce down to the Einstein–Fierz–Pauli theory that we employed.

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