Intrinsic spectral geometry of the Kerr-Newman

JOURNAL OF MATHEMATICAL PHYSICS 47, 033503 共2006兲

Intrinsic spectral geometry of the Kerr-Newman event horizon Martin Engmana兲 and Ricardo Cordero-Soto Departamento de Cencias y Tecnología, Universidad Metropolitana, San Juan, PR 00928, Puerto Rico 共Received 1 November 2005; accepted 18 January 2006; published online 9 March 2006兲

We uniquely and explicitly reconstruct the instantaneous intrinsic metric of the Kerr-Newman event horizon from the spectrum of its Laplacian. In the process we find that the angular momentum parameter, radius, area; and in the uncharged case, mass, can be written in terms of these eigenvalues. In the uncharged case this immediately leads to the unique and explicit determination of the Kerr metric in terms of the spectrum of the event horizon. Robinson’s “no hair” theorem now yields the corollary: One can “hear the shape” of noncharged stationary axially symmetric black hole space-times by listening to the vibrational frequencies of its event horizon only. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2174290兴

I. INTRODUCTION

Although in general the spectrum of the Laplacian on a manifold determines, via the heat kernel, a sequence of invariants which restrict the geometry, it is a rare occasion indeed that the metric is uniquely determined by the eigenvalues 共see, for example, Ref. 6兲. In fact one of the few uniqueness results of this kind was proved by Brüning and Heintze:1 An S1 invariant twodimensional surface diffeomorphic to the sphere, which has, in addition, a mirror symmetry about its equator is uniquely determined by its spectrum. This has been generalized by Zelditch,16 but the Brüning and Heintze result is the appropriate setting for our purposes since this is exactly the class of metrics which include Kerr-Newman event horizons. In 1973 Smarr11 studied the metric of the horizon in terms of a scale parameter and a distortion parameter and in a particularly convenient coordinate system for calculating, for example, the curvature. More recently, the first author of the present paper has used a similar coordinate system to study the spectrum of S1 invariant surfaces. As it turns out, Smarr’s form of the horizon metric is a scaled version of a special case of our form of the metric. S1 invariant metrics on S2 can be described in terms of a single function f共x兲. One can show that the sum of the reciprocals of the nonzero, S1 invariant eigenvalues 共i.e., the trace of the S1 invariant Greens operator兲 is given by an integral involving the function f共x兲. The key to the results we obtain here is that, in the case of the Kerr-Newman event horizon, the function f共x兲 is a simple rational function and the above-mentioned integral can be calculated explicitly. Together with a similar equivariant trace formula, one can now write Smarr’s parameters in terms of these traces and hence explictly display the metric in terms of it’s spectrum. As far as we know, this is the only nontrivial example of this phenomonen 共the Brüning and Heintze result is not constructive in this sense兲. An interesting byproduct of our results 共in the uncharged case兲, together with Robinson’s uniqueness theorem, is the unique reconstruction of space-time in terms of these eigenvalues. We conclude the paper with a discussion of possible physical interpretations of these results.

a兲

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J. Math. Phys. 47, 033503 共2006兲

M. Engman and R. Cordero-Soto

II. SMARR’S FORM OF THE METRIC

The Kerr-Newman metric describing the geometry of a rotating charged black hole written in Kerr ingoing coordinates 共v , r , ␪ , ␾兲 is

冉冉

ds2 = − 1 − +

冊

2mr − e2 共2mr − e2兲2a sin2 ␪ dv d␾ − 2a sin2 ␪ dr d␾ + ⌺ d␪2 dv2 + 2 dr dv − ⌺ ⌺

冊

共r2 + a2兲2 − ␦a2 sin2 ␪ sin2 ␪ d␾2 , ⌺

共1兲

in which 共m , a , e兲 represent, respectively, the total mass, angular momentum per unit mass, and the charge. Also ⌺ = r2 + a2 cos2 ␪ and ␦ = r2 − 2mr + a2 + e2 共we use the lower case letter here, instead of the traditional upper case, to avoid confusion with the equally traditional use of ⌬ for the Laplacian兲. The uncharged 共e = 0兲 case is the Kerr metric. This family of metrics is quite general. To quote Wald15 “¼ the Kerr-Newman family of solutions completely describes all the stationary black holes which can possibly occur in general relativity.” This fact is due to the famous uniquess theorems of Israel,7 Carter and Robinson,10 and Mazur,8 for example. For the Kerr-Newman metric the surface of the event horizon can be thought of as a spacelike slice through the null hypersurface defined by the largest root, r+, of ␦ = 0, i.e., r+ = m + 冑m2 − a2 − e2. The intrinsic instantaneous metric on the event horizon is obtained by setting r = r+, so that dr = 0, and also setting dv = 0 to get 2 = 共r+2 + a2 cos2 ␪兲d␪2 + dseh

冉

共r+2 + a2兲2 r+2 + a2 cos2 ␪

冊

sin2 ␪ d␾2 .

共2兲

In Ref. 11, Smarr defines the scale parameter by ␩ = 冑r+2 + a2 and the distortion parameter by ␤ = a / 冑r+2 + a2 and introducing a new variable x = cos ␪ one finds that the event horizon metric is 2 = ␩2 dseh

冉

冊

1 dx2 + f共x兲d␾2 , f共x兲

共3兲

1 − x2 . 1 − ␤2共1 − x2兲

共4兲

where 共x , ␾兲苸共−1 , 1兲 ⫻ 关0 , 2␲兲 and f共x兲 =

It is well known that the Gauss curvature of a metric in this form is simply K共x兲 = −1 / 共2␩2兲f ⬙共x兲 so that in this case K共x兲 =

1

␩2

冉

冊

1 − ␤2共1 + 3x2兲 , 共1 − ␤2共1 − x2兲兲3

共5兲

and the surface area of the metric is A = 4␲␩2. We point out that in case a = 0 共␤ = 0兲, e = 0 共1兲 gives the Schwarzschild black hole and 共3兲 is the metric of the constant Gauss curvature= 1 / ␩2 metric on S2. III. SPECTRUM OF S1 INVARIANT METRICS

For any Riemannian manifold with metric gij the Laplacian is given by ⌬g = −

1 ⳵

冑g ⳵ x i

冉冑

ggij

冊

⳵ . ⳵x j

This is the Riemannian version of the Klein-Gordon, or D’Alembertian, or wave operator usually denoted by 䊐.


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J. Math. Phys. 47, 033503 共2006兲

Event horizon spectrum

In this section we outline some previous work on the spectrum of the Laplacian on S1 invariant metrics on S2. The interested reader may consult Refs. 2–4 for further details. To simplify the discussion the area of the metric is normalized to A = 4␲ for this section only. The metrics we study have the form dl2 =

1 dx2 + f共x兲d␾2 , f共x兲

共6兲

where 共x , ␾兲苸共−1 , 1兲 ⫻ 关0 , 2␲兲 and f共x兲 satisfies f共−1兲 = 0 = f共1兲 and f ⬘共−1兲 = 2 = −f ⬘共1兲. In this form, it is easy to see that the Gauss curvature of this metric is given by K共x兲 = 共−1 / 2兲f ⬙共x兲. The canonical 共i.e., constant curvature兲 metric is obtained by taking f共x兲 = 1 − x2 and the metric 共3兲 is a homothety 共scaling兲 of a particular example of the general form 共6兲. The Laplacian for the metric 共6兲 is ⌬dl2 = −

冉冊

1 ⳵2 ⳵ ⳵ f共x兲 − . ⳵x ⳵x f共x兲 ⳵␾2

Let ␭ be any eigenvalue of −⌬. We will use the symbols E␭ and dim E␭ to denote the eigenspace for ␭ and it’s multiplicity 共degeneracy兲, respectively. In this paper the symbol ␭m will always mean the mth distinct eigenvalue. We adopt the convention ␭0 = 0. Since S1 共parametrized here by 0 艋 ␾ ⬍ 2␲兲 acts on 共M , g兲 by isometries we can separate variables and because dim E␭m 艋 2m + 1 共see Ref. 4 for the proof兲, the orthogonal decomposition of E␭m has the special form

in which Wk共=W−k兲 is the “eigenspace” 共it might contain only 0兲 of the ordinary differential operator Lk = −

冉冊

d d k2 f共x兲 + dx dx f共x兲

with suitable boundary conditions. It should be observed that dim Wk 艋 1, a value of zero for this dimension occurring when ␭m is not in the spectrum of Lk. The set of positive eigenvalues is given by Spec共dl2兲 = 艛k苸Z Spec Lk and consequently the nonzero part of the spectrum of −⌬ can be studied via the spectra Spec Lk = 兵0 ⬍ ␭1k ⬍ ␭2k ⬍ ¯ ⬍ ␭kj ⬍ ¯ 其 " k 苸 Z. The eigenvalues ␭0j in the case k = 0 above are called the S1 invariant eigenvalues since their eigenfunctions are invariant under the action of the S1 isometry group. If k ⫽ 0 the eigenvalues are called k equivariant or simply of type k ⫽ 0. Each Lk has a Green’s operator, ⌫k : 共H0共M兲兲⬜ → L2共M兲, whose spectrum is 兵1 / ␭kj其⬁j=1, and whose trace is defined by ⬁

␥k ⬅ 兺 j=1

1 ␭kj

.

共7兲

The formulas of present interest were derived in Refs. 2 and 3 and are given by

␥0 =

1 2

冕

1

−1

1 − x2 dx f共x兲

共8兲

if k ⫽ 0.

共9兲

and

␥k =

1 兩k兩

Remark: One must be careful with the definition of ␥0 since ␭00 = 0 is an S1 invariant eigen-


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J. Math. Phys. 47, 033503 共2006兲


value of −⌬. To avoid this difficulty we studied the S1 invariant spectrum of the Laplacian on 1-forms in Ref. 3 and then observed that the nonzero eigenvalues are the same for functions and 1-forms. IV. SPECTRAL DETERMINATION OF THE EVENT HORIZON 2 In case f共x兲 is given by 共4兲 the metric 共3兲 is related to 共6兲 via the homothety dseh = ␩2 dl2, and it is well known that

␭ 苸 Spec共dl2兲

if and only if

␭

␩2

苸 Spec共␩2 dl2兲

so that, after an elementary integration, the trace formulas for the event horizon are

冉

␥0 = ␩2 1 −

2␤2 3

冊

共10兲

and

␥k =

␩2 兩k兩

" k ⫽ 0.

共11兲

For the k = 1 case, for example, one can invert the resulting pair of equations to get

␤2 =

冉冊

3 ␥0 1− 2 ␥1

共12兲

and

␩2 = ␥1 ,

共13兲

and thereby write the event horizon metric explicitly and uniquely in terms of the spectrum as follows. Proposition: With ␥0 and ␥1 defined as in 共7兲 the instantaneous intrinsic metric of the KerrNewman event horizon is given by

2 = ␥1 dseh

冢

1−

冉冊

3 ␥0 1− 共1 − x2兲 2 ␥1 dx2 + 1 − x2

冣

1 − x2 d␾2 . 3 ␥0 1− 1− 共1 − x2兲 2 ␥1

冉冊

共14兲

An immediate consequence of 共11兲 is that the area of the metric has a representation for each k 苸 N given by A = 4␲k␥k .

共15兲

Remarks: 共i兲 One might view Eqs. 共10兲 and 共11兲 as nothing more than definitions of new parameters in terms of the old. On the other hand, the quantities ␥k are fundamental and naturally defined quantities coming from the discrete set of vibrational wave frequencies on the surface. Alternatively, one can think of each trace, ␥k as the sum of the squares of all wavelengths of given quantum number k. 共ii兲 One can use ␥0 together with ␥k for any k to reconstruct the metric. In some sense one can “construct the metric in ways.” 共iii兲 Brüning and Heintze proved that for S1 invariant metrics symmetric about the equator the 1 S invariant spectrum determines the metric. Their result requires the prescription of all of the eigenvalues of the k = 0 spectrum to uniquely determine the surface of revolution. In the example of the present paper the metric is, therefore, uniquely determined once we have knowledge of the


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J. Math. Phys. 47, 033503 共2006兲

Event horizon spectrum

entire list of S1 invariant eigenvalues. For the explicit construction one exchanges complete knowledge of the k = 0 spectrum for partial spectral data, namely the traces of the Greens operators for k = 0 and any k ⫽ 0. All the physical parameters can also be written in terms of the spectra as follows: a2 = 23 共␥1 − ␥0兲,

共16兲

r+2 =

3␥0 − ␥1 , 2

共17兲

m2 =

共 ␥ 1 + e 2兲 2 , 6␥0 − 2␥1

共18兲

as well as the angular velocity and surface gravity. We observe, however, that the mass depends on the charge e as well as the spectrum so that the mass is uniquely determined by the spectrum only in the Kerr 共e = 0兲 case. We set together Remark 共iii兲 with 共16兲 and 共18兲共with e = 0兲, 共1兲, and Robinson’s uniqueness theorem10 to observe. Theorem: A noncharged stationary axially symmetric asymptotically flat vacuum space-time with a regular event-horizon is uniquely determined by the S1 invariant spectrum of the intrinsically defined Laplacian on its event horizon. The space-time is explicitly constructed from the S1 invariant trace together with any k ⫽ 0 trace of the associated Green’s operator on the event horizon. V. DISCUSSION

It is well known 共see Refs. 5, 13, and 9, among many others兲 that the quasinormal mode frequencies for scalar, electromagnetic, and gravitational radiation of black holes are related to the separation constants arising from separating variables in the Teukolsky master equation. These “characteristic sounds” of the black hole are considered to be observable in an astrophysical sense. The angular operator 共and, respectively, its separation constants兲 coming from the separation of variables is closely related to the Laplacian 共and, respectively, its eigenvalues兲 on the event horizon in the sense that, for scalar fields, they both reduce to the Laplacian 共and corresponding eigenvalues兲 on the constant curvature S2 for the Schwarzschild 共a = 0兲 case. There is also some evidence 共this is a work in progress兲 that the a ⬎ 0 Teukolsky angular equation can be viewed, after a change of variables, as a Laplace eigenvalue equation for a fourth order Taylor polynomial approximation of the horizon metric function. We hope to pursue these matters in our future work. On a more philosophical note, the reader may have noticed that the main result of this paper is consistent with the holographic principle 共Refs. 12 and 14兲 in as much as the structure of the 共3 + 1 dimensional兲 Kerr-Newman space-time is encoded in the intrinsic spectral data of the 共twodimensional兲 event horizon surface. These phenomena suggest that the spectrum of the Laplacian on event horizons is playing an important, if rather subtle and not well understood, role in the physics of the space-time manifold. ACKNOWLEDGMENTS

The authors thank Thomas Powell, Steve Zelditch, and Roberto Ramírez for their ideas and assistance in the preparation of this paper. This work was partially supported by the NSF Grant: Model Institutes for Excellence at UMET. Brüning, J., and Heintze, E., “Spektrale starrheit gewisser drehflächen,” Math. Ann. 269, 95–101 共1992兲. Engman, M., “New spectral characterization theorems for S2,” Pac. J. Math. 154, 215–229 共1992兲. 3 Engman, M., “Trace formulae for S1 invariant Green’s operators on S2,” Manuscr. Math. 93, 357–368 共1997兲. 4 Engman, M., “Sharp bounds for eigenvalues and multiplicities on surfaces of revolution,” Pac. J. Math. 186, 29–37 共1998兲. 1 2


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