The Kerr Memory Effect at Null Infinity

Bondi, van der Burg, Metzner, and Sachs bondi1960gravitational ; Bondi:1962px ; cit:sachs1 wanted to define a concept of asymptotic flatness at null infinity. The falloffs needed to be restrictive enough that unphyiscal spacetimes — such as those with infinite energy — would be ruled out, yet not so restrictive such that physical spacetimes and gravitational waves would be ruled out. While the falloffs may differ in the literature, we will use the choice made by BMS bondi1960gravitational ; Bondi:1962px ; cit:sachs1 ; stromingernotes :

The class of allowed asymptotic line elements for these falloffs is given by

where $\Theta^{A}\in\{\theta,\phi\}$ and the uppercase Latin indices run over $\theta,\phi$. $D_{A}$ is the covariant derivative on the 2–sphere with respect to the 2–sphere metric, $\gamma_{AB}$. The function $m_{\text{bondi}}$ is the Bondi mass aspect, which is in general a function of $u$ and the angles, $\theta\,,\phi$. This can be used to obtain the entire Bondi mass after integrating $m_{\text{bondi}}$ over the entire 2–sphere at null infinity. In the case of the Kerr spacetime, the Bondi mass is simply, $M$, the mass of the black hole. $N_{A}$ is the angular moment aspect. Contracting $N_{A}$ with the generator of rotations and integrating over the entire sphere is related to the total angular momentum of the spacetime. $C_{AB}$ is another field which is symmetric and traceless ($\gamma^{AB}C_{AB}=0$). The retarded time derivative of $C_{AB}$ is in fact the Bondi news tensor,

The news tensor is the gravitational analogue of the Maxwell field strength and its square is proportional to the energy flux across $\mathcal{I}^{+}$ stromingernotes ; compere_lectures .

It is important to note that $N_{A}$ is defined slightly differently in various parts of the literature — usually depending on the asymptotic expansion (2). For instance, Comperé in refs Compere_final_state ; 45offinalstate has a decomposition that leads to $N_{A}$ being defined as⁴⁴4This corresponds to pure vacuum and the mass contribution is omitted.

However, Strominger in ref stromingernotes and Comperé in ref compere_lectures uses the decomposition (2), which leads to $N_{A}$ being defined as

Here, $g^{(1)}_{uA}$ corresponds to the $r^{-1}$ expansion in $g_{uA}$. We will opt to use the second definition as the superrotation charge will not be changed. This can be seen explicitly in the case of the Kerr solution.

Symmetries of a spacetime are associated to Killing vectors of that spacetime. Hence, before discussing the charges associated with symmetries in our spacetime, we must first briefly discuss the Killing vectors of our spacetime - or rather, the asymptotic Killing vectors. The most general Killing vector, $\boldsymbol{\xi}$, that preserves the metric (2) to leading order is stromingernotes

Here, $f$ is a function of that angular coordinates $(\theta,\phi)$ only and $D^{2}$ is the standard Laplacian on the 2–sphere. However, since the analysis conducted here is only a calculation to linear order — as done by HPS HPSrot , the Killing vector is truncated:

This does indeed beg the question of whether a second order analysis would still show that the memory effect — as seen from null infinity — is still a supertranslation with the Killing vector (6).

It is well known that the symmetries of spacetime are associated to conserved charges via Noether’s theorem. Before the discovery of the BMS group, the largest symmetry group was the Poincaré group, which has associated conserved charges such as energy and momentum. The BMS group, which is an infinite dimensional group of symmetries at null infinity also has charges associated to supertranslations and superrotations.

Supertranslation charge and its conservation is given by⁵⁵5Note that integration is carried out over the 2–sphere at null infinity. stromingernotes

Here $f$ is a function of angular coordinates⁶⁶6Note that while $f$ is often chosen to be a spherical harmonic function, it is not limited to be only a spherical harmonic. and can be thought of as the generator of supertranslations. In general, the supertranslation charges will depends on advanced/retarded time. This is simply due to $m_{\text{bondi}}$ — in general — depending on advanced/retarded time. In the case where $f=1$ the conserved charge is energy while in the case where $f$ is a $l=1$ spherical harmonic function we have conservation of ADM momentum.

Superrotation charge and its conservation is given by⁷⁷7This definition of superrotation charge follows from using (5) as our definition of $N_{A}$. In the case that one uses (4), superrotation charge is expressed via Barnich_troes $$Q_{Y}=\frac{1}{16\pi}\int\mathrm{d}^{2}\Theta\,\sqrt{\gamma}\,Y^{A}\Big{[}2N_{A}+\frac{1}{16}\partial_{A}\Big{(}C_{BC}C^{BC}\Big{)}\Big{]}\,.$$

where $Y^{A}$ is an arbitrary vector field on the 2–sphere ⁸⁸8As is the case for $f$, the components of $Y^{A}$ are often chosen to be spherical harmonic functions, however, the components of $Y^{A}$ are not limited to be spherical harmonics.. In the case that $Y^{A}$ is one of the 6 global conformal Killing vectors on the 2–sphere (9) expresses conservation of ADM angular momentum and boost charges.

The notation used here indicates that supertransformation charge at future null infinity, $\mathcal{I}^{+}$ should match the supertransformation charge at past null infinity, $\mathcal{I}^{-}$. Furthermore, these matching conditions are a statement about having a well-posed scattering problem in general relativity. The existence of these conserved charges is — at least in principle — verifiable with the gravitational memory effect, which may be detectable in the near future.

During the last decade there has been a lot of discussion regarding the gravitational memory effect and the direct correspondence of this effect with the BMS group — for instance, one may see refs stromingernotes ; HPSrot ; HPS ; compere_lectures ; Compere_final_state ; donnaysuperathorizon ; extendedsymm ; Wald_bHM . It was shown by HPS HPSrot that a deformation due to a shockwave defined via an energy-momentum tensor is equivalent to a BMS supertranslation at null infinity (for a Schwarzschild black hole). Such a supertranslation is given by taking the Lie derivative of the spacetime metric — in this case, the Schwarzschild metric — along the asymptotic Killing vector⁹⁹9Note the sign change here due to changing from retarded time to advanced time.,

However, as HPS HPSrot state, supertranslations only equate to part of the deformation a gravitational wave would produce when striking a black hole.

Intuitively, gravitational waves, which carry energy, should impart some of this energy to the black hole it strikes and it should alter the mass and/or (angular) momentum. In fact, it was shown by HPS HPSrot that there was a change in the mass of a Schwarzschild black hole due to the monopole (denoted by $\mu$ in ref HPSrot ) contribution of the shockwave. Therefore, the full¹⁰¹⁰10The word full is used here, however, it is worth noting that this is only an analysis at linear order. “memory effect” due to such a shockwave was written as¹¹¹¹11The Heaviside terms here have been neglected as we are assuming the shockwave has already struck the black hole.

Here $\delta g_{\mu\nu}$ refers to the permanent change in the spacetime due to a gravitational wave. Therefore, the mass of the hairy Schwarzschild black hole would be $m=M+\mu$ after the gravitational shockwave strikes the black hole. However, while the mass of the black hole may change, the Bondi mass of the black hole — at least at linear order — does not change. This further emphasises that the memory effect is not entirely captured by BMS supertransformations. However, the memory effect as seen from null infinity is entirely captured by BMS supertranslations.

This discussion is important when analysing the calculations in this paper. Applying the same gravitational shockwave used in HPSrot ; stromingernotes ; cit:bhm to the Kerr solution is a non-trivial task and thus one does not know for certain what the other components to the memory effect will appear. A common suspicion within the community is that there may be another monopole contribution to the mass term that appears in the $g_{u\phi}$ component of the metric. This would imply a change in the angular momentum of the Kerr black hole due to a gravitational shockwave — which will not be seen at null infinity by observation of the superrotation charge.

As discussed by HPS HPSrot ; stromingernotes supertranslation hair do not impart supertranslation charge. Recall that the supertranslation charge observed at future null infinity is defined as follows:

It is now clear that since the Bondi mass aspect, $m_{\text{bondi}}$, is not changed due to the supertranslation alone we will not have any supertranslation charge turned on by supertranslation hair. However it is worth noting that in Donnay et al. cit:bhm and ref KKSofthair an analysis of the Schwarzschild and Kaluza–Klein spacetimes in the near horizon limit is conducted. This analysis shows that nontrivial supertranslation charge is turned on at the horizon due to the gravitational wave that is absent at null infinity. Furthermore, one may see ref nearhorizon_req for an analysis relating near horizon displacement effects to supertransformation charges.

Supertranslation hair does however, carry superrotation charge. The superrotation charge that is measured at future null infinity is given by

Using (20) we see that superrotation charge that is present at null infinity is

and

Here the first lines correspond to the bald Kerr black hole superrotation charges and can easily be recovered if the supertranslation function $f$ vanishes. Furthermore, as one would expect when $a=0$ we recover the superrotation charges of the hairy Schwarzschild black hole HPSrot ; cit:bhm ; KKSofthair . As shown by Barnich and Troessaert in Barnich_troes , when $Y^{\phi}$ is the Killing vector, $\partial_{\phi}$, (25) corresponds to conservation of angular momentum (for the bald Kerr black hole). For the Hairy Kerr black hole, one may see that the zero-mode superrotation charge (when $f=0$ and $Y^{\phi}=1$) given by (25), does not change and will still correspond to conservation of angular momentum.

Detection of these charges still remains an open question. One first requires an operational notion of finite infinity. Secondly, we require a notion of what higher order charges would be observed as in our detectors. In theory, the supertranslation field should be detectable via classical tests of general relativity such as the bending of light Compere_final_state ; Teo_lensing ; stdetection_req . However, as Comperé discusses in Compere_final_state , one would need an array of detectors surrounding the central object in order to deduce the superrotation charges, thereby confirming the existence of the supertranslation field.

In subsection 1.4 we discussed that it has been shown the mass, $M$ — in the case of the Schwarzschild spacetime — is changed by a factor of $\mu$ which is the monopole contribution to the shockwave. We discussed that in the case of Kerr, this shockwave may also change the mass term present in the $g_{uA}$ terms. This would change the angular momentum of the Kerr black hole, as one may expect from the passing of a gravitational wave. Since the mass of the black hole is changed, one may ask why there is no supertranslation charge found at null infinity or why the zero-mode of superrotation charge (angular momentum) is not changed.

Recall, however, that supertranslation and superrotation charges are only defined for $\mathcal{I}^{+}$ and $\mathcal{I}^{-}$. This, therefore, becomes a statement of what an observer at null infinity measures as the memory effect rather than what the memory effect may be for all observers. Indeed, it seems that the change in mass and, therefore, changes in momentum and/or angular momentum are not measurable at null infinity via the measurement of superrotation charges.