Local Invariants Vanishing on Stationary Horizons: A Diagnostic for Locating Black Holes

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by  Don N. Page

Local Invariants Vanishing on Stationary Horizons:
A Diagnostic for Locating Black Holes
Don N. Page∗ and Andrey A. Shoom†
Theoretical Physics Institute, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
(Dated: 2015 March 19)
Inspired by the example of Abdelqader and Lake for the Kerr metric, we construct local scalar
polynomialcurvatureinvariantsthatvanishonthehorizonofanystationaryblackhole: thesquared
norms of the wedge products of n linearly independent gradients of scalar polynomial curvature
invariants, where n is thelocal cohomogeneity of the spacetime.
PACSnumbers: 04.70.Bw,04.50.Gh,04.20.Cv,02.40.Hw Alberta-Thy-1-15,arXiv:1501.03510
5
1
0 How may one locate a black hole? This question may of a generic stationary spacetime, at a generic point in
2 be quite challengingevenfor numericalcomputation, for the spacetime outside an horizon these gradients will be
the location of a black hole horizon is a delicate issue. linearly independent and spacelike, but on the horizon,
r
a Here we propose a method which generically gives the a linear combination will become null. This implies that
M preciselocationofanystationaryhorizonandshouldgive the squared norm of the wedge product of the gradients
the approximate location of a nearly stationary horizon. will vanish on a stationary horizon.

9 Hence it should be useful for numerical computations in OurproceduredoesnotassumetheEinsteinequations,
1
general relativity. butonlyasmoothstationaryhorizon(anullhypersurface

For a general black hole spacetime, the location of its generated by a Killing vector field that is timelike out-
]
c surface (the event horizon, which is the boundary of the side the horizon but null on the horizon). Therefore, in
q
regionfromwhichcausalcurvescangotoasymptoticfu- principletheinvariantstobeusedintheprocedurecould
-
r ture null infinity) depends on the future evolution of the includethosefromtheRiccitensorRαβ,suchastheRicci
g spacetime and is not determined locally. However, for scalar R = Rα or RαβR . However, because the main
[ α αβ
black holes that settle down to become stationary, one interest may be in Ricci-flat (Rαβ = 0) spacetimes that

4 might ask whether one can find a local invariant that is solvethevacuumEinsteinequations,intheexamplesbe-
v generically nonzero off the horizon but vanishes on the low we shall consider only invariants obtained from the
0 horizon. Inparticular,onemightlookforascalarpolyno- Riemann curvature tensor R (or from the Weyl ten-
αβγδ
1
mialcurvatureinvariant[1–7], whichis ascalarobtained sor C , which is the trace-free part of the Riemann
5 αβγδ
by complete contraction of all the indices of a polyno- tensor) that need not vanish even when R =0.
3 αβ
0 mial in the Riemann curvature tensor Rαβγδ and its co- In the original (30 Dec. 2014) version of their paper,
. variant derivatives. For example, Anders Karlhede, Ulf Abdelqader and Lake [9] gave the following six curva-
01 Lindstr¨omandJanAman[8]showedthatRαβγδ;ǫRαβγδ;ǫ ture invariants for the Kerr metric, which we here copy
crosses zero and switches sign as one crosses the horizon directly from that paper:
5
1 oftheSchwarzschildmetric,andonecaneasilyshowthat
v: thisisalsotrueforanysmoothstaticsphericallysymmet- I1 ≡Cαβγδ Cαβγδ , (1)
ric horizon. An invariant that vanishes on more general
i
X stationaryhorizonscouldbeusefulinnumericalrelativity
r for finding the approximate location of the horizon once I2 ≡C∗αβγδ Cαβγδ , (2)
a thespacetimehassettleddowntobecomeapproximately
stationary.
Majd Abdelqader and Kayll Lake [9] have recently I ≡∇ C ∇µCαβγδ , (3)
3 µ αβγδ
found a local scalar polynomial curvature invariant that
vanishes on the horizon of the Kerr black hole. After
casting this invariantinto a simpler form that is propor- I ≡∇ C ∇µC∗αβγδ , (4)
4 µ αβγδ
tional to the squared norm of the wedge product of two
curvature-invariantgradients,werealizedthattheproce-
dure generalizes to give a way to locate any nonsingular I ≡k kµ , (5)
5 µ
stationaryhorizonin terms oflocalcurvature invariants.
Essentially, if one constructs as many gradients of inde- and
pendent curvature invariants as the local cohomogeneity
I ≡l lµ , (6)
6 µ
where C is the Weyl tensor, C∗ its dual, k ≡
αβγδ αβγδ µ
∗ [email protected] −∇ I , and l ≡ −∇ I . Then they showed that in
µ 1 µ µ 2
† [email protected] the Kerr metric the dimensionless scalar nonpolynomial
2
curvatureinvariant(withscalarpolynomialcurvaturein- of the spacetime (the codimension of the maximal di-
variant numerator) mensional orbits of the isometry group of the local met-
ric, ignoring the breaking of any of these local isome-
(I5+I6)2−(12/5)2 I12+I22 I32+I42 tries by global considerations, such as the way that the
Q ≡ (cid:0) (cid:1)(cid:0) (cid:1) (7)
2 108 I 2+I 2 5/2 identifications of flat space to make a torus break the
1 2
(cid:0) (cid:1) local rotational part of the isometry group), so that at
vanishes on the black hole horizon. generic points outside an horizon of a generic metric of
We foundthatthe syzygy(inthis contextafunctional local cohomogeneity n, the squared norm of the wedge
relationshipbetweeninvariantsforaparticularspacetime productwouldbe positive. (We use the metric signature
geometry) that Abdelqader and Lake [9] discovered for −++···.) Let us now prove that this squared norm
the Kerr metric, that vanishes on a stationary horizon.
Let D be the dimensionality of the spacetime, and let
12
I −I + (I I −I I )=0, (8) m be the maximal dimension of the orbits of the local
6 5 1 3 2 4
5
isometry group. Then n = D−m is the local cohomo-
may be expressedas the realpart of the complex syzygy geneity of the spacetime. For a stationary spacetime of
local cohomogeneity n with an event horizon, let ξµ de-
12
∇ (I +iI )∇µ(I +iI )= (I +iI )(I +iI ) note the Killing vector field that on the horizon is its
µ 1 2 1 2 1 2 3 4
5 null generator, which is orthogonal to the null horizon
(9)
hypersurface. In a neighborhood outside the horizon, ξµ
whoseimaginarypartgivesasyzygyforaseventhinvari-
willbetimelike,withξµξ <0inourchoiceofsignature.
ant, µ
Let {S(i)}, for i ranging from 1 to n, denote a set of n
6 functionally independent nonconstant scalar polynomial
I ≡k lµ = (I I +I I ) . (10)
7 µ 5 1 4 2 3 curvatureinvariants(totalscalarcontractionsoverallin-
dicesofpolynomialsofcurvaturetensorsandoftheir co-
Using both the real and imaginary parts of this complex
variantderivatives). For example, some or allof the S(i)
syzygy, one may readily see that
couldbechosenfromthesetI ...I givenabove(though
1 7
27 I 2+I 2 5/2Q =(k kµ)(l lν)−(k lµ)(l kν), remembering that I2, I4, I6, and I7 above are only de-
(cid:0) 1 2 (cid:1) 2 µ ν µ ν fined for D = 4), but one could also choose invariants
(11)
fromtotalscalarcontractionsofotherpolynomialsinthe
whichisthesquarednormofthewedgeproductdI ∧dI
1 2 Riemann curvature tensor and its covariant derivatives.
of the gradients of the Kretschmann invariant I and of
1 Thenlet{dS(i)}bethesetofexteriorderivatives(gra-
theChern-PontryagininvariantI (foraRicci-flatspace-
2
dients) of the n scalar polynomial curvature invariants,
time; otherwise in four dimensions the Kretschmann in-
variantisRαβγδRαβγδ =I1+2RαβRαβ−(1/3)R2,though with components S;(µi) = ∇µS(i). Since the curvature
the Chern-Pontryagin invariant R∗ Rαβγδ = I has invariants are invariant under translations by the lo-
αβγδ 2
no correction from the Ricci tensor). cal isometry group, their gradients all lie with the n-
Although it is incidental to the main point of our pa- dimensional local cohomogeneity part of the cotangent
per,wealsofoundanothersyzygyfromaminorextension space at each point [6]. The wedge product of n gradi-
of the work of Abdelqader and Lake [9], ents, the n-form W =dS(1)∧dS(2)∧···∧dS(n), will be
proportional to the volume form in this n-dimensional
ℑ (I +iI )4(I −iI )3 =0, (12) part of the cotangent space at each point of spacetime,
h 1 2 3 4 i
with a proportionality depending upon the spacetime
where ℑ denotes the imaginary part of what follows. point. At generic points in a generic spacetime, the pro-
(The real part is equal to the absolute value in this case portionalitywill be nonzerowherethe gradientsof the n
and does not vanish to give yet another syzygy.) With scalar invariants are linearly independent, but there can
these three syzygies for the seven real scalar polynomial beasetofpoints(genericallyhypersurfaces)wherethen
curvatureinvariants,I ,...,I ,oneisleftwithfourinde- gradientsarenotlinearlyindependent,sothatthewedge
1 7
pendent invariants, precisely the right number to deter- product vanishes there and the proportionality is zero.
mine generically (up to eight sign choices) the values of The Hodge dual ∗W is then an m-form that at
the two Kerr parameters M and a and of the two nonig- each point of the spacetime lies entirely within the m-
norablecoordinatesr andθ. (Noneofthesethreeremain dimensional part of the cotangent space that is gener-
syzygies when one adds a cosmological constant, so the ated by the local isometries. It will be proportional to
syzygies for that case remain to be found.) the m-dimensional volume element of that part of the
Eq. (11) for the Abdelqader-Lake invariant Q that cotangent space, though the location-dependent propor-
2
theyfoundvanishesonthehorizonoftheKerrmetricin- tionality will be zeroat the same set of spacetime points
spiredtherealizationthatthesquarednormofthewedge at which the wedge product W vanishes.
product of n gradients of independent local smooth cur- On a stationary horizon, which is a null hypersurface
vatureinvariantswouldvanishonthehorizonofanysta- generatedbytheKillingvectorξµ thatistimelikeoutside
tionary black hole, where n is the local cohomogeneity thehorizonbutnullonthe horizon(andhence bothpar-
3
allel and orthogonal to the horizon), the m-dimensional functionallyindependentscalarpolynomialcurvaturein-
part of the cotangent space will include the null one- variants so that one can choose n of them in such a way
form, with one-form components ξ = g ξν, which is thatthesquarednormofthewedgeproductoftheirgra-
µ µν
metrically equivalentto the Killing vectorξµ that is null dientsisgenericallynonvanishingawayfromthehorizon?
and hypersurface-orthogonal at the horizon, as well as If the answer is ‘no,’ then although our theorem above
other spacelike one-forms metrically equivalent to other wouldremaintrue,the squarednormofthe wedgeprod-
Killing vectors at the horizon that are all spacelike if uctofthegradientsofanynscalarpolynomialcurvature
m>1. Therefore, this m-dimensionalpart of the cotan- invariantswouldvanishinawholeneighborhoodofasta-
gent space will be null on the stationary horizon, and tionaryhorizon,soitwouldnotbeusefulforlocatingthe
hence ∗W will also be null, having zero squared norm. horizon. Based upon the work of Alan Coley, Sigbjørn
Since the squared norm of W, appropriately defined, is Hervik, and Nicos Pelavas [3–5], one might conjecture
thesameasthatofitsHodgedual∗W,thesquarednorm that the answer to the question is ‘yes,’ but so far we do
ofthewedgeproductW ofnscalarpolynomialcurvature not have a rigorous proof.
invariants will be zero on a stationary horizon. Let us now consider various examples in four-
There is of course in this argument the implicit as- dimensionalspacetimesofthefactthatthesquarednorm
sumptionthatthe spacetimeissufficientlyregularatthe of the wedge product of n gradients of curvature invari-
event horizon that the gradients of the n scalar polyno- ants vanishes at the horizon of a stationary black hole.
mialcurvatureinvariantsarewell-definedatthe horizon, First, consider a spherically symmetric static black
so thattheir wedge productW is well-definedandfinite. hole, so the codimension is n = 1 (the radial direction,
We summarize this result by the following theorem: say with coordinate r). Then any smooth curvature in-
variant,suchastheKretschmanninvariantR Rαβγδ,
αβγδ
Theorem. Foraspacetimeoflocalcohomogeneitynthat which is the same as I for a Ricci-flat spacetime, will
1
contains a stationary horizon (a nullhypersurface that is have a gradientin the radial directionthat becomes null
orthogonal to a Killing vector field that is null there and on the horizon (e.g., as seen in a frame parallely propa-
hence lies within the hypersurface and is its null gener- gatedbyaninfallinggeodesic),sothesquareofthenorm
ator) and which has n scalar polynomial curvature in- of the gradient of the Kretschmann invariant (with only
variants S(i) whose gradients are well-defined there, the one gradient in the wedge product in this n = 1 case)
n-form wedge product W =dS(1)∧dS(2)∧···∧dS(n) has will be zero on any spherically symmetric static black
zero squared norm on the horizon, hole horizon.
Second, consider a stationary axisymmetricblack hole
1
kWk2 ≡ δα1...αngβ1γ1···gβnγn infourdimensions,whichhastwocommutingKillingvec-
n! β1...βn
tor fields and cohomogeneity n = 2, such as the Kerr
×S;(α11)···S;(αnn)S;(γ11)···S;(γnn) =0. (13) metric. Then we need the wedge product of two gradi-
ents for its squared norm to be positive at generic loca-
where the permutation tensor δα1...αn is +1 if α ...α
β1...βn 1 n tions outside the black hole but vanishing on the hori-
is an even permutation of β ...β , is −1 if α ...α is
1 n 1 n zon. For a rotating black hole such as Kerr, one can
an odd permutation of β ...β , and is zero otherwise
1 n take the squared norm of the wedge product dI ∧dI
(including all cases of repeated indices upstairs or down- 1 2
as Abdelqader and Lake [9] effectively did (though origi-
stairs).
nally without realizing explicitly that their invariant Q
2
Then kWk2 is itself a scalar polynomial curvature in- is proportional to the squared norm of this wedge prod-
uct before we discovered and pointed out this fact). For
variant that vanishes on any stationary horizon smooth
a static axisymmetric black hole, the invariants I , I ,
enoughforW tobewell-definedthere. Ofcourse,forthis 2 4
I , and I vanish, so one could instead take the squared
tobeusefulforlocatingastationaryhorizon,thenscalar 6 7
polynomialcurvatureinvariantsS(i) shouldbe chosento normofdI1∧dI3,ofdI1∧dI5,ofdI3∧dI5,orofanyother
be functionally independent so that kWk2 is positive at pair of independent nonvanishing scalar polynomial cur-
vature invariants, such as the trace of higher powers of
genericpointsinthespacetimeoutsidethehorizon. This
will not always be possible, such as for the cosmological the curvature tensor like RαβγδRγδǫζRǫζαβ.
horizon of de Sitter spacetime, which is totally homoge- In these cases with more than one Killing vector that
neous and hence has cohomogeneity n = 0, though of all commute (so that the dimensionality m of the isome-
coursefor this spacetime there arecosmologicalhorizons try group is the same as the number of Killing vectors),
running through every point. thesquarednormofthewedgeproductofnindependent
scalar polynomial curvature invariants will also vanish
This raises the following question: For a spacetime of
local cohomogeneity n with a Killing horizon that is a
locally unique hypersurface1, are there always enough
includes points not on a Killing horizon, unlike the case for de
Sitter or certain other examples, such as spacetimes having a
covariantlyconstant nullvector field,thathaveKillinghorizons
1 Bylocallyunique,wemeanthatanyneighborhoodofthehorizon throughallpoints.
4
at the fixed points of the Killing vectors other than the Kerr-(A)dS spacetime has a reflection symmetry about
one that becomes null on the horizon, such as on the the equatorial plane hypersurface, if one ignores the
axes of axisymmetric spacetimes. In such cases we do change in sign of the volume element given by the to-
not know how to construct scalar polynomial curvature tallyantisymmetricLevi-Civitatensor. Inthis case,gra-
invariants that vanish only on the horizon, so that will dients of scalar polynomial curvature invariants that do
be left as a challenge for the future, as well as the chal- notinvolveoddpowersoftheLevi-Civitatensor(e.g.,not
lenge as to whether for specific spacetimes such as Kerr, counting I , I , and I in D = 4) will lie in the (n−1)-
2 4 7
there is a scalar polynomial curvature invariant without dimensionallocalcohomogeneitypartofthehypersurface
any covariant derivatives of the Riemann curvature ten- ofreflectionsymmetry,sothat the wedgeproductofany
sorthatvanishes onthe horizonorthatvanishesonly on n such gradients will vanish there. When the NUT pa-
the horizon. rameter is nonzero, there is no reflection symmetry, but
Third, consider a distorted static black hole in four generically there remain distorted hypersurfaces where
dimensions that has no spatial Killing vector fields on the wedge product vanishes.
and outside the horizon, so the local cohomogeneity is We found that there are generically other hypersur-
n = 3. Then we need the squared norm of the wedge faces away from the horizons and axes on which the
productofthreegradientsofscalarpolynomialcurvature wedge product of n = 2 gradients vanishes. How-
invariants, such as dI ∧ dI ∧ dI or wedge products ever, we found that if we took the three scalar polyno-
1 3 5
using the gradients of the traces of higher powers of the mial curvature invariants S(1) = R Rαβγδ, S(2) =
αβγδ
curvature tensor and/or of its covariantderivatives. Rαβ Rγδ Rǫζ , and S(3) = R Rαβγδ;ǫ, then we
γδ ǫζ αβ αβγδ;ǫ
There are of course analogous examples in spacetimes
havethree pairsofwedgeproducts, W =dS(1)∧dS(2),
12
ofhigherdimensionD >4. Ifthey arestatic andspheri- W = dS(2) ∧ dS(3), and W = dS(3) ∧ dS(1), that
23 31
callysymmetric,thenforanydimensionD ≥4,thelocal
eachvanishoncertaincurvesinthe(r,θ)planeandwith
cohomogeneity is n = 1, and the gradient of any scalar
each pair concurrently vanishing at certain intersection
polynomialcurvatureinvariant(suchastheKretschmann
points in the (r,θ) plane, but with no triple intersec-
invariant) that has a nonvanishing gradient just outside
tionswhereallthreewedgeproductsconcurrentlyvanish
the horizonwill haveits squarednormbecoming zero on
for the sample values of M, a, and L that we chose.
the horizon.
Therefore, it appears that if one takes the sum of the
The general Kerr-NUT-(A)dS metric in spacetime di-
three squared norms of the wedge products each multi-
mension D [10] has local cohomogeneity n=⌊D/2⌋, the
pliedbypositivecoefficients,thenforgenericparameters
greatestintegernotgreaterthanD/2. Thatis,theD =4
this will be nonzero everywhere except at the horizons
casehastwocommutingKillingvectorfieldsandlocalco-
and fixed points of the isometry. However, we did find
homogeneityn=2(asthespecialcaseoftheKerrmetric
that for certain choices of the Kerr-NUT-(A)dS parame-
doeswithboththeNUTparameterandthecosmological
ters, even this sum of squares vanishes at certain sets of
constant zero). Analogously, the D = 5 case has three
points in the spacetime away from the stationary hori-
commuting Killing vector fields (one becoming null on
zons and fixed points of the isometries. We believe this
the horizon and two axial ones) and also n=⌊5/2⌋=2.
vanishinganywherecouldbeeliminatedforanysetofthe
Then D = 6 and D = 7 have n = 3, D = 8 and D = 9
threeparametersbychoosingasumofsixsquarednorms
have n = 4, D = 10 and D = 11 have n = 5, etc. In
ofthe wedge products of the six pairs ofa suitable setof
each case the squared norm of the wedge product of the
four gradients of scalar polynomial curvature invariants.
gradientsofnscalarpolynomialcurvatureinvariantswill
One generalprocedure for calculating a scalarpolyno-
vanish on the horizons (as well as on the axes surfaces,
mial curvature invariant I , vanishing on a stationary
the fixed points of the axial Killing vector fields). (p)
horizon,thatshouldgenericallybe positive whereverthe
One might conjecture that one could take the traces
Killing vectors span the maximal dimension m of the lo-
of the 2nd through the (n+1)th powers of the Riemann
cal isometry group that includes a timelike direction (so
curvature tensor as one choice of n invariants. We have
that one is not at a stationary horizon or fixed point of
looked at sample values of the 3 parameters (mass M,
the isometry, and so that the n independent directions
rotation parameter a, and NUT parameter L at fixed
of the local cohomogeneity are all spacelike) would be
cosmological constant) of the Kerr-NUT-(A)dS metric
to take a set of n + p scalar polynomial curvature in-
in spacetime dimension D = 4 and have found that in- variants, {S(i)} for i running from 1 to n+p with suf-
deed the square of the norm of the wedge product of the
ficiently large p. (For example, p could be the number
traces of the 2nd and 3rd powers of the Riemann curva-
of parameters for a certain class of spacetimes, such as
turetensor(theKretschmanninvariantR Rαβγδ and
αβγδ p=D−1fortheKerr-NUT-(A)dSfamilyofmetrics[10]
Rαβ Rγδ Rǫζ ) is generically nonzeroawayfrom the of even dimension D, or p=D−2 for odd D, for which
γδ ǫζ αβ
horizonsandaxes,buttherearecurvesinthe(r,θ)plane there are n+p algebraically independent scalar polyno-
(hypersurfacesofthefullspacetimethatareneitherhori- mialcurvatureinvariants.) Thenthereare(n+p)!/(n!p!)
zonsnoraxes)wherethesquareofthenormofthewedge ways of choosing n gradients to form the squared norm
product vanishes. of wedge product n-form, and one can multiply each by
In particular, when the NUT parameter L is zero, the theproductofthesquarednormsoftheremaininggradi-
5
ents andthen sum overthe (n+p)!/(n!p!) choices to get ‘curvature-invariant quasi-horizons’ where the squared
an invariant I that will generically be positive where norm of the appropriate wedge product of gradients of
(p)
all the gradients are spacelike, as they generically would scalar polynomial curvature invariants vanishes, as ana-
beundertheconditionsabove. Ofcourse,preciselywhat logues to apparent horizons, though in the nonstation-
scalarpolynomialcurvatureinvariantI onegetswould ary case the locations of the ‘curvature-invariant quasi-
(p)
depend upon which set of n+p scalar polynomial cur- horizons’ may well depend on what scalar polynomial
vature invariants {S(i)} one chooses, and some special curvature invariants are chosenfor the wedge product of
choices could still conceivably give I = 0 even away their gradients.
(p)
from stationary horizons and fixed points of the isome-
try group. We are grateful to Majd Abdelqader for explaining
In conclusion, we have found a way to generalize the some aspects of the work of Abdelqader and Lake [9]
results of Abdelqader and Lake [9] for the Kerr metric and for confirming that the new syzygy of Eq. 12 is in-
to give ways of constructing scalar polynomialcurvature deedasyzygyoftheKerrmetric,whichwehavealsonow
invariants that vanish on any stationary horizon. These checked along with the others. We also appreciate help-
invariantsmightbe usefulfornumericallyestimatingthe ful advice from Alan Coley, Sigbjørn Hervik, and Eric
locationofhorizonsonceaspacetime settles downtobe- Woolgar for revisions of the manuscript. This research
coming nearly stationary. They may also be interest- has been supported in part by the Natural Sciences and
ing in nonstationary black hole spacetimes for defining Engineering Research Council of Canada.
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