Let us turn now to gravity and change the focus slightly. While many of the results found in gauge theories hold, we would like to investigate some effects specific to gravity at this point. The reader may have noticed that we omitted the Rarita–Schwinger field which also has a gauge symmetry.
We shall cover supergravity and this fermionic field in particular in a subsequent publication. We should note that the historical context of [41] was the question of how to define energy in general relativity [43].
In this section we investigate the Noether two-form that we receive from Noether’s second theorem when used on the diffeomorphism symmetry of a linear metric perturbation in Einstein gravity. Interestingly, the two-form Noether charge for diffeomorphisms is well known in the literature [44,49,60,69] and has been used for a wide variety of problems.16 It is connected to the entropy of a black hole [60] and has been connected with information theoretic measures [72].
It has been used to derive the soft graviton theorem [5,9,11,13]. The fact that it can be derived with the help of Noether’s second theorem, however, has not been sufficiently emphasized in the literature.
Starting with the Einstein–Hilbert action with Gibbons–Hawking–York boundary term S=SEH+SGHY SEH= 1
2κ2 Z
M
ddx√−gR SGHY= 1 κ2
Z
∂M
dd−1x√−γK, (94) whereγis the metric on ∂M, one may follow the procedure outlined in Sec. 2 and then write Ward identities as in 3. There is an additional complication, beyond the treatment there, in that the action depends on two derivatives of the metric. This leads to a∇µ∇µρterm in (7). One might consider avoiding this issue by switching to a first-order formulation; however, the extra term affects neither the Ward identity nor the two-formk, assuming the boundary term is consistent with the variational principle. One finds the two-form in [49]:
kµν= 1
2κ2(∇µξν−∇νξµ). (95)
In order to avoid several complications, it is convenient to work with linearized gravity instead of the full nonlinear theory. Indeed, we don’t know how to work with the path integral of the nonlinear theory, in any case.
Assuming a perturbationhµν=gµν−gµνwhere we usegµνas a classical vacuum-like back-ground andgµν a metric that satisfies the nonlinear equations of motion of gravity with some matter energy-momentum tensorTµν
Rµν− 1
2gµνR=κ2Tµν, (96)
the equations of motion for the perturbation are given by Eµν=
∇µ∇νh+∇λ∇λhµν−2∇λ∇(µhν)λ−gµν(∇λ∇λh−∇λ∇ρhλρ)
. (97)
More generally, a linearized equation of motion for gravity coupled to matter is given by Eµν= 1
2κ2
RLµν−1
2gµνRL
= (Tµν+tµν) =Tµν (98) where superscriptLindicates linearized quantities. We also includedtµν, which are the higher order terms of the expansion of the Einstein tensor.
Using Noether’s second theorem
2Eµν∇µξν=∇µ(2Eµνξν) =∇µSµ(E,ξ) (99) and the currentjµ(which can be derived from a rather tedious calculation), we find the two-form
kµν= 1 2κ2
ξρ∇σHρσνµ+ 1
2Hρσνµ∇ρξσ
(100) with
Hµανβ= 1 2
hανgµβ+hµβgαν−hαβgµν−hµνgαβ
(101) andhµνis the trace reversed metric. We also could have perturbed eq. (95) for this expression.
The same expression may be found in [44] and was obtained originally in [73]. We excluded the energy-momentum tensorTµνfrom the calculation because it will not appear in kµνξ . The quantityHµναβis known as thesuperpotentialin the literature and it enjoys the same symmetries as the Riemann tensor Rµναβ. Again, the leading order of this two-form is linearin the field
16See also e.g., [70,71].
hµν, suggesting that the linear perturbation to the metric field behaves like a Goldstone boson in the sense explained above. One could have included a cosmological constant in the action; the two-formkwould not have been changed.
Let us mention that for the case of a Killing vector of the background
∇µξiν+∇νξiµ=0 (102)
whereienumerates the number of Killing vector fields, one finds thatkµνξ is the quantity tra-ditionally used to calculate conserved charges at spacial infinityi0 (letkbe a tensor density by multiplying with√−g)
Mi= Z
σ0
ki. (103)
Crucially, then,
jµ=∇νkµν=Tµνξν. (104)
Note that we do not need to prove that the conserved quantity associated with the Killing vectors are codimension two surface integrals because it is manifestly the divergence of a two-form by Noether’s second theorem.
There is a general path integral formalism available for metric perturbations of Einstein gravity [74] which we use here to make contact with the soft theorem [11]at tree level. For the large dif-feomorphisms in asymptotically flat space, or equivalently asymptotic Killing vectors satisfying
∇(µξν) = O(1/r)atI, the formalism developed in Sec. 3 together with [11] implies choosingρ such that we are integrating the two-form over the surfaceσ(a sphere) atI−+
Q[ξ] = Z
σ
?kξ (105)
this integral can then be turned into an integral over all of17I where Q(ξ) =
Z
I ?(?d?kξ) (106)
= 1 2κ2
Z
I dSν
2κ2Tµρξρ+∇µ(Hνρσµ+1
2Hρσνµ)∇ρξσ+1
2Hνµρσ(∇µ∇ρξσ−Rκµρσξκ) We used once again kξ = √
−g k. In the second line, we pulled out the currentSµusing the linearized Einstein equations in the form (see e.g. [73])
2κ2Tµν=∇α∇βHµναβ+1
2RνραβHµραβ. (107)
Clearly, this result is more involved than the case whenξis a Killing vector. In the Killing vector case, one can use that∇α∇βξρ = Rλαβρξλ to make the last term vanish. For asymptotically flat space times, this term is still very suppressed at asymptotic infinity since the background Riemann tensorRµναβbehaves at least asO(r−3).
We can now use Sec. 3 with the path integral for “quantum” gravity, insert our result from above and retrieve Weinberg’s soft theorem in the case of an asymptotically flat manifold. The procedure is very similar to what was presented in Ssec. 4.1. One first chooses Bondi coordinates and specializesξto BMS supertranslations. For these, the leading term at the boundary isξv=T, whereT is an arbitrary function of the coordinates of the sphere at infinity.
To make contact with [11], we note that we can use Sec. 3 with the path integral for “quantum”
gravity [74], insert our result from above and retrieve Weinberg’s soft theorem in the case of an asymptotically flat manifold. The procedure is very similar to what was presented in Ssec. 4.1.
One first chooses Bondi coordinates where
gµνdxµ⊗dxν= −du2−2dudr+2r2γzz¯dzdz¯ (108) hµνdxµ⊗dxν=2mB
r du2−2Uzdudz−2Uz¯dudz¯+rCzzdz2+rCz¯z¯dz¯2 (109) and specializesξto BMS supertranslations
ξµ∂µ=T ∂u−1
r(Dz¯T ∂z¯+DzT ∂z) +DzDzT ∂r+o(r−1) (110)
17Note, that we are assuming good behavior when approaching timelike infinity alongIin the spirit of [11].
whereT = T(z, ¯z). With these coordinates, the only component ofHµνκλappearing in Q(ξ) = δR,ξSEHisHurur = −huugrr = −2mrB. For the BMS transformations, the leading term at the boundary isξu=Tsuch that we end up with
Q(ξ) = 1 κ2
Z
σ
dzdz γ¯ zz¯T mB (111)
which we recognize as the BMS charge defined in [11]. We could also have started from (106) to get to the result integrated over all ofI±. As before, we may insert this into the path integral for a set of insertionsΦiby the method described in 3. The resulting identity takes the form of (42) after rewritingQ(ξ)as an integral over all ofI. We recognize the Ward identity as the leading soft graviton theorem.
5 Discussion
Noether’s second theorem, is an old but somewhat underappreciated tool, which acquires new significance in light of recent developments. By combining the robustness of the path integral for-malism with the elegance of Noether’s second theorem we find that writing down Ward identities for residual gauge symmetries becomes essentially automatic. We have focused on gauge symme-tries for fields that obey bosonic statistics, but we will treat the case of the Rarita–Schwinger field as the gravitino in supergravity in an upcoming publication. There a host of new and interesting effects appears. Much like in the case of the BMS symmetry, which contains Poincaré transfor-mations, we find that the enhanced symmetry contains the usual supersymmetry charges of the supergravity background. The resulting Ward identities form the soft theorem of the gravitino.
Our approach sheds some light on interesting connections that emerge from the literature. Es-pecially important is the interpretation of some residual gauge symmetries as a shift symmetry in the spirit of Goldstone’s theorem, as well as the connection between Strominger et al’s highly influential soft graviton theorem as a consequence of BMS symmetry and Wald’s important dis-covery that the Noether two-form integrated over the bifurcation two-sphere is the entropy of the black hole.
Additionally, the current investigation opens the path to the study of anomalies for residual gauge symmetries. By anomaly, we mean actual one-loop effects connected to the path integral measure as opposed to classical effects like shifts of the boundary conditions of the path integral, which lead to classical central charges. We know that the subleading soft theorem in gravity as well as the leading and subleading soft theorem in YM theory get corrected at the first loop level from the study of scattering amplitudes. We conclude that some of the residual gauge symmetries cannot survive quantization. This is not a problem, since they are not traditional gauge symme-tries; by definition they are not gauge fixed out of the path integral measure. An anomaly in these symmetries, thus, does not signal the breakdown of the theory.
More speculatively, we alluded several times to the possibility of slicing open the path integral and writing Ward identities for subregions of the total spacetime. It is desirable to explore how that works in detail, and its physical consequences. One may also consider transformations in-volving the Fadeev–Popov ghosts, which ultimately should lead to statements in the BRST and BV formalisms.
Acknowledgments BUWS would like to thank Andrew Strominger for useful discussions as well as the Department of Physics at Brown U. for continued hospitality. BUWS is supported by the Cheng Yu-Tung fund, the Center for Mathematical Sciences and Applications at Harvard Uni-versity, and NSF grant 1205550. SGA is grateful for correspondence with Nemani Suryanarayana, and for discussions with Antal Jevicki and Samir Mathur. SGA is supported by US DOE grant de-sc0010010. The authors are thankful to Matteo Rosso and Bruno Le Floch for useful comments.
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