Gravitational lensing in Brill spacetimes Mourad Hallaaand Volker Perlickb ZARM University of Bremen 28359 Bremen Germany

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Gravitational lensing in Brill spacetimes

Mourad Hallaa) and Volker Perlickb)

ZARM, University of Bremen, 28359 Bremen, Germany

(Dated: 2 February 2023)

We consider the Brill metric which is an electrovacuum solution to Einstein’s ﬁeld

equation. It depends on three parameters, a mass parameter m, a NUT parameter

land a charge parameter e. If the charge parameter is small, the metric describes a

black hole; if it is suﬃciently big, it describes a wormhole. We determine the relevant

lensing features both in the black-hole and in the wormhole case. In particular, we

give formulas for the photon spheres, for the angular radius of the shadow and for

the deﬂection angle. We illustrate the lensing features with the help of an eﬀective

potential and in terms of embedding diagrams. To that end we make use of the

fact that each lightlike geodesic is contained in a (coordinate) cone and that it is a

geodesic of a Riemannian optical metric on this cone. By the Gauss-Bonnet theorem,

the sign of the Gaussian curvature of the optical metric determines the sign of the

deﬂection angle. In the wormhole case the deﬂection angle may be negative which

means that light rays are repelled from the center.

Keywords: Brill wormhole, Brill black hole, gravitational lensing, photon circle,

Gauss-Bonnet theorem, Gaussian curvature, optical metric, embedding diagram

a)Electronic mail: mourad.halla@zarm.uni-bremen.de.

b)Electronic mail: perlick@zarm.uni-bremen.de.

arXiv:2210.01611v3 [gr-qc] 31 Jan 2023

I. INTRODUCTION

Gravitational lensing is one of the most important tools for observing (ultra-)compact ob-

jects such as black holes or wormholes. In this case the weak-ﬁeld small-angle approximation

that is often employed in lensing is not applicable because light rays can make arbitrarily

many turns around the central object. Then one has to use the full spacetime formalism of

general relativity, without approximation, for determining the lensing features, see e.g. the

living review by Perlick1.

In this paper we want to apply this formalism to lensing in the Brill spacetime which

is an exact solution to the Einstein-Maxwell equations, found by Brill2in 1964. The Brill

metric depends on a mass parameter m, a NUT parameter land a charge parameter e. For

l= 0 the metric reduces to the Reissner-Nordstr¨om metric which is static and spherically

symmetric. As the light rays in the Reissner-Nordstr¨om metric have been extensively dis-

cussed (see e.g. Chadrasekhar3), we will restrict our investigation to the case l6= 0. Then

the metric is still stationary and it still admits an SO(3,R) symmetry; however, it is no

longer static and it is not spherically symmetric in the usual sense because the orbits of

the SO(3,R) symmetry are 3-dimensional timelike hypersurfaces rather than 2-dimensional

spacelike spheres. For e= 0 the Brill metric reduces to the Newman-Unti-Tamburino (NUT)

vacuum solution4which describes a black hole. For non-zero e, the Brill metric still describes

a black hole (now with charge) as long as e2is small, but for suﬃciently big e2it describes

a traversable wormhole, see Cl´ement et al.6. These Brill wormholes are the only known

traversable wormhole solutions to Einstein’s ﬁeld equation (in 4 spacetime dimensions) with

an energy-momentum tensor that satisﬁes all energy conditions. We believe that for this

reason it is worthwhile to study their lensing features in detail.

The paper is organized as follows. In Sec. II we review the basic features of Brill

spacetimes. In Sec. III and Sec. IV we derive, respectively, the relevant equations for

general geodesics and for lightlike geodesics. We discuss the lensing features of Brill black

holes in Sec. Vand of Brill wormholes in Sec. VI.

II. BRILL SPACETIMES

The Brill metric, also known as the Reissner-Nordstr¨om-NUT metric, is an exact solution

of the Einstein-Maxwell equations that was found by Brill in 19642as a generalization of

the NUT metric4. It depends on three parameters, m,land e. In Boyer-Lindquist-type

coordinates (t, r, ϑ, ϕ) the Brill metric reads

gµν dxµdxν=−(r−m)2+b

r2+l2dt −2l(cos ϑ+C)dϕ2

+(r2+l2)dr2

(r−m)2+b+ (r2+l2)dϑ2+ sin2ϑ dϕ2,

(1)

with

b:= e2−m2−l2.(2)

ϑand ϕare the standard coordinates on the two-sphere S2, whereas the time coordinate t

and the radial coordinate rrange over all of R, unless in the case l= 0 where the radial

coordinate has to be restricted to r > 0, or r < 0, because there is a curvature singularity

at r= 0.

The metric is stationary, but not static, on the domain where gtt <0. Moreover, it admits

an SO(3,R) symmetry that will be discussed below. However, as the orbits of the SO(3,R)

action are not two-dimensional spacelike spheres, the metric is not spherically symmetric in

the usual sense of the word.

In (1) we have written the metric in a way that involves, in addition to the three param-

eters m,land ealso another, dimensionless, parameter Cwhich was not included in the

original work of Brill. It was introduced only later by Manko and Ruiz5for the NUT metric

and it generalizes naturally to the Brill metric. By a coordinate transformation

t0=t−2lCϕ, r0=r, ϕ0=ϕ, ϑ0=ϑ(3)

one can transform the Manko-Ruiz parameter Cto zero near any one point oﬀ the axis, so

the local geometry oﬀ the axis is unaﬀected by changing C. On the axis, there is a conic

singularity, if l6= 0, and this singularity is inﬂuenced by C: For C= 1 the singularity is on

the upper half axis (ϑ= 0), for C=−1 it is on the lower half-axis (ϑ=π), and for any other

value of Cit is on both half-axes, symmetrically distributed for C= 0 and asymmetrically

for other values of C. Each of the three parameters m,eand lhas the dimension of a

length. mis the mass parameter which will be assumed non-negative throughout, m≥0.

e2=q2+p2is the combination of an electric charge parameter qand a magnetic charge

parameter p; obviously e2≥0. lis the gravitomagnetic charge, also known as the NUT

parameter, which may take any value l∈]−∞,+∞[. In the analogy between gravitation and

electromagnetism, mcorresponds to the electric charge, while lcorresponds to a magnetic

(monopole) charge.

For l= 0, the metric reduces to the Reissner-Nordstr¨om metric. Then there is a curvature

singularity at r= 0, so we have to restrict to the region r > 0 (or to the region r < 0). If m >

0, the spacetime region r > 0 describes a black hole, with horizons at r±=m±√m2−e2,

for e2≤m2and a naked singularity for e2> m2; the ﬁrst case includes of course the

Schwarzschild metric with e2= 0. If m= 0, we have a massless naked singularity for

e2>0 and ﬂat Minkowski spacetime for e2= 0. As the Reissner-Nordstr¨om metric has been

extensively covered in the literature, we exclude the case l= 0 in the rest of this paper.

With l6= 0 the Brill metric describes a black hole for b≤0 and a traversable wormhole

for b > 0, see Cl´ement et al.6. In the case of black holes, there are two horizons at r±=

m±√m2+l2−e2. For b= 0, i.e. q2+p2=m2+l2, it was shown by Cl´ement et al.6

that the metric is a special case of the Israel-Wilson-Perj`es metric; in this case we have a

black hole with a degenerate horizon. The two special cases m= 0 and e2= 0 are included:

m= 0 gives us a massless black hole for l2≥e2and a massless wormhole for l2< e2.

e2= 0 gives us the NUT metric4. The NUT metric is a solution to Einstein’s vacuum ﬁeld

equation that describes a black hole, with horizons at r±=m±√m2+l2. The region

between the two horizons is isometric to a cosmological vacuum solution found by Taub7;

therefore, the analytic extension of the NUT solution beyond the outer horizon is properly

called the Taub-NUT solution.

In the black-hole case, for gravitational lensing it is reasonable to restrict rto the domain

of outer communication, i.e., to the region outside of the outer horizon m+√m2+l2−e2<

r < ∞. Clearly, an observer in the domain of outer communication can receive only light

signals that are completely contained in the domain of outer communication, so as long as

we do not consider observers who are foolhardy enough to jump into the black hole the

region beyond the outer horizon is of no relevance. In the wormhole case, however, there

are no horizons which means that any observer can receive signals from the entire domain

−∞ < r < ∞.

We have said that we assume that truns over all of Rand that then for l6= 0 there is a

conic singularity on the axis. This singularity can actually be removed by making the time

coordinate periodic, with the period 4π|lC|, as was suggested by Misner8(for the uncharged

NUT metric with C=−1). This, however, leads to a closed timelike curve through each

event where ∂tis timelike, i.e., to a most drastic kind of causality violation, so we will not

follow this suggestion. It is true that also without making the time coordinate periodic

there are closed timelike curves in the Brill (or in particular NUT) spacetime, but if the

NUT parameter is suﬃciently small they are restricted to an arbitrarily small region near

the axis, so one may argue that this does not lead to any pathological behaviour that is

actually observable.

In the rest of this paper, a Brill spacetime with m≥0 and l6= 0 will be assumed. For

b≤0, we limit ourselves to the domain of outer communication of the black hole, whereas

in the wormhole case b > 0 we have to consider the entire domain −∞ < r < ∞. It was

emphasized already by Cl´ement et al.6that Brill wormholes are traversable, i.e., that luminal

and subluminal signals can travel from r=−∞ to r=∞and vice versa. This distinguishes

Brill wormholes from the Einstein-Rosen bridge9. As the Brill metric is a solution to the

Einstein-Maxwell equations, there is no exotic matter involved. This dinstinguishes the Brill

wormholes from the Teo wormholes10 (which include the Morris-Thorne wormholes11) and

also from a class of wormholes considered by Halla and Perlick12 that is closely related to

but not identical with the class of Teo wormholes. All these wormholes are traversable but

by Einstein’s ﬁeld equation they have negative energy densities near the throat. The Brill

wormholes do not violate any of the energy conditions (weak, strong or dominant). The price

we have to pay is in the weaker asymptotic structure: Whereas Teo wormholes have two ends

which are asymptotically ﬂat in the sense that the metric approaches the Minkowski metric,

the Brill wormhole metrics are asymptotically ﬂat only in the sense that the curvature goes

to zero for r→ ±∞; however, the spheres (r= const., t = const.) do not become spacelike

surfaces with area 4πr2for big r. Also, there is the above-mentioned conical singularity on

the axis. Nonetheless, some readers may ﬁnd it attractive to have wormhole solutions to

Einstein’s ﬁeld equation without exotic matter, even if they have some other pathologies.

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GravitationallensinginBrillspacetimesMouradHallaa)andVolkerPerlickb)ZARM,UniversityofBremen,28359Bremen,Germany(Dated:2February2023)WeconsidertheBrillmetricwhichisanelectrovacuumsolutiontoEinstein'seldequation.Itdependsonthreeparameters,amassparameterm,aNUTparameterlandachargeparametere.Ifthecharge...

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Gravitational lensing in Brill spacetimes Mourad Hallaaand Volker Perlickb ZARM University of Bremen 28359 Bremen Germany

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