Pedro F. Gonzalez-Dıaz- Observable effects from spacetime tunneling

8/3/2019 Pedro F. Gonzalez-Daz- Observable effects from spacetime tunneling

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arXiv:g

r-qc/9708044v119Aug1997

Observable effects from spacetime tunneling

Pedro F. Gonzalez-DazCentro de Fsica Miguel Catalan, Instituto de Matematicas y Fsica Fundamental,

Consejo Superior de Investigaciones Cientficas, Serrano 121, 28006 Madrid (SPAIN)(August 12, 1997)

Assuming that spacetime tunnels -wormholes and ringholes- naturally exist in the universe, weinvestigate the conditions making them embeddible in Friedmann space, and the possible observableeffects of these tunnels, including: lensing and frequency-shifting of emitting sources, discontinuouschange of background temperature, broadening and intensity enhancement of spectral lines, so as adramatic increase of the luminosity of any object at the tunnels throat.

PACS number(s): 04.20.Cv, 98.80.Es, 98.80.Hw

I. INTRODUCTION

Solutions to Einstein equations corresponding to

spacetimes with closed timelike curves (CTCs) havestirred the relativistics, following developments by Lanc-zos [1], van Stockum [2], Godel [3], Misner [4] and, morerecently, Morris and Thorne [5], Gott [6] and Jensen andSoleng [7] The reason for the excitement and subsequentgeneral disbelief resided much in that, being consistentsolutions to Einstein equations for very special kinds ofmatter, the proposed spacetimes allow for the possibil-ity of time travel and, hence for potential violations ofcausality [8]. Here we look at the idea that, rather thanbeing interpreted as constructs to be eventually built upfrom future highly developed technology, spacetime tun-nels generating CTCs may spontaneously exist in some

regions of our universe, and give rise to observable effectswhich could be detected even with present technology.

Thus, we consider CTCs generated along spacetimetunnels whose mouths embed in distant regions with theFriedmann geometry of the overall universe. The neces-sary condition for such tunnels to occur in a given regionis that, in that region there is a certain proportion ofmatter with negative energy [8,9]. Two tunnel topolo-gies have been considered so far. That of a two-spherewhich gives rise to wormholes [5,10], and that of a two-torus which is associated with the so-called ringholes [11].Both tunneling types are traversable and convertible intotimemachines generating CTCs by simply letting one of

the holes mouths to move toward the other [5,10,11], butwhereas the energy density is everywhere negative nearthe throat of a wormhole, it still becomes positive forvalues of the angle 2 (defining the position on the sur-face circles determined by the torus sections) such that2 h > 2 > h, with h = arccos ba , where a andb are the radius of the circumference generated by thecircular axis of the torus and that of a torus section, re-spectively, at the ringholes throat [11].

The purpose of the present work is to investigate underwhat conditions can a tunnel be embedded in a cosmo-logical spacetime, and explore the effects that the inner

properties of spacetime tunnels may have on the observ-able characteristics of astronomical objects placed be-yond, or passing through these tunnels, relative to an

observer whose line of sight to the object traverses ordoes not traverse the given tunnel. Most of the emphasiswill be placed on ringholes, but the results will be alwayscompared with those expected from wormholes.

II. EMBEDDING A TUNNEL IN FRIEDMANN

SPACE

Assuming that the amount of negative mass equalsthat of positive mass, the gravitational field that rep-resents tunneling through a traversable ringhole can bedescribed by a spacetime metric [11]:

ds2 = dt2 + ( nr

)2dl2 + m2d21 + b2d22, (2.1)

where < t < , l is the proper radial distance ofeach transversal section of the ringholes torus, 2 > h, the ringhole wouldbehave like the wormhole; i.e. it acts as a diverging lensnear the throat. This property may be crucial to identifythe existence of these holes in the universe and distin-guish between them.

Metric (2.1) can easily be converted into that of aspherical wormhole by applying the coordinate change[11] a 0, 2 + 2 , 1 , and allowing agiven purely negative mass M to induce a general fac-tor e2(r,M) in the gtt components of the metric tensor.

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For a section of constant 1 and 2, the ringhole metric(2.1) can be written as

ds2 = b2(d2 + d2), (2.3)where

d = dtb

, d = ndlrb

. (2.4)

In the approximation in which b b0 (where b0 is thevalue ofb at the throat), l b [11], one can integrate thesecond of expressions (2.4) to

= 0

+ ln

r

a+

b

a cos 2

ab

+r

b cos 2

cos2. (2.5)

When a null geodesic is considered, ds = 0 and we obtain

b =db

d= b

db

dt=

rl

n

d = d,

or integrating

= + 0, (2.6)and hence, following the same line of reasoning as in e.g.Ref. [12], we obtain that the frequency of the light raysis either red-shifted while traveling from mouth to throat

by an amount

z = 0

0= b

b= rl

nb, (2.7)

or blue-shifted by exactly the same amount if the lightrays travel from throat to mouth. Therefore, along com-plete passage through a static ringhole, the frequencyof a light ray preserves its initial value. However, theembedding of the ringhole mouths in two respectivelydistant regions of a Friedmann spacetime with scale fac-tor R(F), conformal time F =

dtFR

and distance fromthe origin RF, ensures that such two distant regions arered-shifted relative to one another by the known cosmo-

logical amount predicted from = 0R(FF)/R(F)[12], both for static ring- and worm-holes. It follows thatthese tunnels cannot embed in the background Fried-mann spacetime, unless for the case that the tunnels beallowed to be no longer static, with one of the holesmouths moving with respect to the other mouth with anonzero velocity, V, such that the Doppler shift result-ing from this motion would equal the usual cosmologicalred-shift:

z = H Dc

= Vc

, (2.8)

with H the Hubbles constant and D the distance, bothfor worm- and ring-hole. In the latter case, the initialdistance between mouths can be calculated to be:

Lr = 2

nb

rldb = 2a, (2.9)

where we have discarded the integration constant, so that(2.8) can finally be re-written as

z = HDc 2Hra

c, (2.10)

with Hr =b

b2.

The metric of a ringhole whose mouths move with ve-locity V relative to one another has the form [11]

ds2 = [1 + glF(l)sin 2]2 dt2 + dl2 + m2d21 + b2d22,(2.11)

where g =

2

(dV/dt), = 1/

1 V

2

, l =

b2

b2

0,with b0 the value of b at the throat, and F(l) is a formfactor that, if we assumes mouth A to be moving, van-ishes in the half of the ringhole with the other mouth, andrises monotonously from 0 to 1 as one moves from thethroat to mouth A. For this ringhole to be embedded inFriedmann spacetime, the velocity V must be restrictedto be

V = HD =R

RD, (2.12)

with the overhead dot meaning derivative respect to theFriedmann time tF. In this case, the embedding of sec-

tions in the ringhole space with constant angles 1 and2 in sections of Friedmann space with constant angles and , implies

dtF = (1 + glF(l)sin (0)2 )dt, (2.13)

with tF and t the times for, respectively, Friedmann andringhole spacetimes.

Let us evaluate g at given constant values of (0)2 and

D. We have

g = 2dV

dt= 2D

dH

dtF

(1 + glF(l)sin

(0)2 ),

where (2.13) has been used. From this we obtain

g =2DH2(q 1)

1 2DH2(q 1)lF(l)sin (0)2, (2.14)

where q = RRR2

is the deceleration parameter at thetime when the ringhole is formed. Hence, we obtain asthe metric of a ringhole embeddible in Friedmann space:

ds2 =

1 DH2(q 1)lF(l)sin (0)2

(1H2D2)22

dt2

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+ dl2 + m2d21 + b2d22. (2.15)

The corresponding embeddible metric for wormholescan again be obtained from this by using the transfor-mation [11] a 0, 2 + 2 , 1 , and introducinga factor e2 in the gtt component of the metric tensor.On the other hand, a ringhole embeddible in Friedmannspace will develop closed timelike curves and hence con-vert into timemachine at sufficiently late times, providedthat the time shift induced by relative motion betweenmouths exceeds the distance between mouths [9], i.e.

tF >

1H2D2

(q 1)HlF(l)Hr sin (0)2

, (2.16)where Eqns. (2.10), (2.13) and (2.14) have been used.Thus, for a given time tF, and provided that there existsa certain proportion of matter with negative energy, aringhole timemachine can be spontaneously created for

the set of parameters l, F(l), Hr and (0)

2which satisfy

(2.16), so that any astronomical object going through theringhole should traverse CTCs in the inner nonchronalregion of the ringhole to travel back- or for-ward in time,depending on whether it enters the ringhole by its movingor stationary mouth [11].

III. OBSERVABLE EFFECTS

A. Lensing

Let us assume that at one of the ringholes mouthsthere is a source of radiation which, since the hole istraversable, can be detected by an observer placed at theother mouth. Some of the rays emitted by the source willcross the ringholes throat on one of the angular horizons,h say, where they will not undergo any deflection [11].Other rays will however cross the throat along values ofthe angle 2 such that 2h > 2 > h and, therefore,will be deflected towards the angular horizon by gravi-tational repulsion of the negative energy placed on thishorizon side. One should then expect that the observerwill detect a double image of the source, such as it isdescribed in Fig. 1. Assuming that the total amount ofnegative mass is M < 0, we then have

= 2 = 4G|M|b( 2)c2 . (3.1)

Besides, for small it can be obtained

=2 h

LNb =

2 hd

2b. (3.2)

From (3.1) and (3.2),

2 (h + ) + (1 + h) = 0, (3.3)where

=b2dRs

, =bdRs

, h =bhdRs

,

with Rs = 2G|M|/c2. The solutions to the quadratic(3.3) are

=

h +

(h

)2

4

2 . (3.4)

It follows that: (i) if h > 2 there will be two raysdeflected to the observer, (ii) if h < 2 deflectionwill fully block all rays from reaching the observer, andfinally (iii) if h = 2 there will be an umbra re-gion surrounded by a caustic where rays accummulateto produce a brilliant region of highly enhanced intensity[13]. Double imaging in ringholes would tend to occur forgenerally small mass |M| and mouth separation d, andgenerally large radius of the throat. The effect would alsobe enhanced when b approaches a.

In the above analysis we have assumed that direct rayslinking the source to the observer pass on the angularhorizon at or near the ringholes throat. Moreover, itcan be shown that if the velocity of the particles comingto the detector is v, then the observer will detect a secondimage of the source with a shifted frequency

=

vG|M|b( 2)c2 , (3.5)

where is the relativistic factor for velocity v, such as ithappens in cosmic strings [14].

The lensing and frequency-shifting effects could not oc-cur in wormholes, where one must assume the negativeenergy to be uniformly or radially distributed in the re-gion surrounding the throat. However, they should hap-pen in ringholes even when the mouths are at rest relativeto one another. The latter is the case which is actuallyassumed in the above calculation and Fig. 1. For anembeddible ringhole, the calculation goes along similarsteps, with = + ( = ) but, since both and should be very small, the final result cannot appreciablydiffer from (3.4).

B. Discontinuous change of temperature

If the ringhole is allowed to have its two mouths inrelative motion at a speed V, setting two objects initially

at rest on one side of the throat, but near to it, one on theangular horizon and the other at an angle 2h > 2 >h, then when the mouth on the other side starts moving,there will appear a transverse velocity component for theobject at 2 h > 2 > h toward the object on theangular horizon due to the diverging effect of negativemass. This velocity component is given by

u = VV ,

with V the relativistic factor for velocity V and asgiven by (3.1). Letting the object on 2h > 2 > h

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be a source of radiation and the object on the angularhorizon h an observer, we deduce that the latter mustdetect a discontinuous change of radiation frequency dueto Doppler shift.

In the cosmological context, the ringhole will be backlitby a uniform black body radiation background, so that

the Doppler shift would result in a discontinuous changeof temperature,

T

T=

VvG|M|b( 2)c2 , (3.6)

originated from the motion of one mouth relative to theother that can make the source and observer traverse thethroat of the ringhole with moving mouths, in a similareffect to that is also induced by cosmic strings [14].

This discontinuous change of temperature would notbe expected to happen in wormholes where the throatregion is filled with negative energy only and thereforethere is no angular horizons.

C. Line broadening

On the other hand, assuming the usual Maxwellianvelocity distribution, the relative probability for atomswith resonant emission frequency 0 [15], entering onemouth of a ringhole with its mouths in relative motionat velocity V, to have the Z-component (we assume Z tobe the main axis of the ringhole) of their velocity withvalues between vZ and vZ + dvZ will be given by

expmA( 2)bc3

2kB

V

V Rs

20

TMW

( 0)2cd

0

, (3.7)

where Eqn. (2.8) has been used, mA is the mass ofthe atom and TMW is the temperature of the microwavebackground. So, the lines of the spectrum will have theirmaximum at w0 and a full Doppler-broadened width athalf their maximum height

40kBVV RsTMW ln 2

mAb( 2)c3 . (3.8)

A large broadening would then be expected for spectrallines from atoms emitting through a ringhole embeddedin Friedmann space, from values of 2 close to , but notthrough any kind of wormholes.

D. Spectral intensity and luminosity

Finally, let us consider the effect that spacetime tun-nels may have in the spectral intensity of distant objects.If a number N of emitting two-level atoms cross a ring-hole, they will pass through some regions with negativeenergy and some regions with positive energy near thethroat. In regions with negative energy chaotically as-sambled, the temperature will be negative and give rise

to population inversion in the energy levels of the atoms,in a direct effect which differs from the pumping by athree-level systems that most simply characterizes lasers[15]. Instead, here the population inversion is originatedby the fact that in the Boltzmanns law

N1 = N2 exp h

kBT

,

(where N1 and N2 are the population of, respectively, thefirst and second atomic levels, assumed to be nondegen-erate) negative values of T necessarily imply N2 > N1.In this case, the beam intensity I

I

Z= +(N2 N1)F()

hB

cn

I, (3.9)

(where [15] F()d is the fraction of transitions in whichthe photon frequency lies in a small range d about fre-quency , B is the Einstein absorption coefficient, n is

the refractive index and is the total volume of the con-sidered region) will grow with distance along the ringhole,through the atomic gas. Meanwhile, in the region withpositive energy, N1 > N2, and hence I must decreasewith distance along direction Z, through the atomic gas.Along a distance Z 2b0 on the line of sight, onewould then expect a large increase of the intensity of thelight coming from the compactified regions of galaxieswhich are confined to a width of the order b0, when thegiven galaxies cross a ringholes throat.

On the other hand, as far as we are dealing with a ring-hole which is embeddible in some region of the universe,the entering of the galaxy in one mouth implies its simul-

taneous exit at the other mouth [11], so that the over-all observable effect will be that of a very brilliant com-pact region of size b0, embedded in the larger galaxy.This effect would be expected for all considered tunnels,though it became most apparent in the case of ringholesbecause of the existence of angular horizons which makequite neater the separation of the brilliant region from adarker background.

IV. PERSPECTIVE EFFECTS

All the effects discussed so far refer to the particu-lar case where the observers line of sight to the objectpasses through the traversable spacetime tunnels. How-ever, direct access by sight to the region surrounding theringholes throat by an observer sustends a given, moreor less deformed, cone of direct observation, generated byrevolution around an axis parallel to the ringhole axis atthe center of a transversal section of the torus, with anangle whose vertex is at the throat. The maximum valueof that angle is expected to be (Fig. 2)

arctan

Lr2a

= 45,

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where (2.9) has been used.Observers whose line of sight to the throat lies out-

side that cone will have only a projected evidence of theobjects passing through the tunnel throat and, in thecase of a static ringhole or in the more realistic case ofa ringhole with its mouths set in relative motion with

nonzero velocity, they would detect the object reach-ing the two mouths, as well as some secondary effectsinduced from the primary effects occurring around thethroat (strong broad emission lines, very high luminosi-ties induced by the smallness of the throat, etc), butnot these primary effects themselves. Among the pos-sible secondary effects one may include the existence ofregions filled with ionized gas which would be observablewhatever angle between the line of sight and the ringholeaxis may be formed. These regions should be originatedfrom the high-intensity beam of ionizing photons possess-ing the whole collection of electronic transition frequen-cies of the atoms and molecules whose electronic energylevels were inversely populated in the throat region.

It is rather a curious fact that the most remarkablefeatures of the emerging overall picture (see Fig. 2) can,at least qualitatively, be accommodated to the currentunified models of galaxies with active nuclei, class 1 andclass 2 Seyferts and quasars [16]. These models attributethe differences among these objects to the presence of adusty torus of dense molecular gas surrounding a blackhole. The dusty torus serves as well to give rise to acone of direct observation with angle of about 45. Justlike for ringholes, all differences can then be attributedto merely being observed by lines of sight which lie insideor outside this cone.

ACKNOWLEDGMENTS

For useful comments, the author thanks M. Moles ofIMAFF. This research was supported by DGICYT underResearch Projects No. PB94-0107 and No. PB93-0139.

[1] K. Lanczos, Z. Phys. 21, 73 (1924).[2] W.J. van Stockum, Proc. R. Soc. Edin. 57, 135 (1937).[3] K. Godel, Rev. Mod. Phys. 21, 447 (1949).[4] C.W. Misner, in Relativity Theory and Astrophysics I.

Relativity and Cosmology, edited by J. Ehlers (AmericanMathematical Society, Providence, RI, 1967).

[5] M.S. Morris and K.S. Thorne, Am. J. Phys. 56, 395(1988).

[6] J. Gott, Phys. Rev. Lett. 66, 1126 (1991).[7] B. Jensen and H.H. Soleng, Phys. Rev. D45, 3528 (1992).[8] K.S. Thorne, Black Holes and Time Warps (Norton, New

York, 1994).

[9] M. Wisser, Lorentzian Wormholes (Am. Inst. Phys.Press, Woodbury, 1996).

[10] M.J. Morris, K.S. Thorne and U. Yurtsever, Phys. Rev.Lett.c 61, 1446 (1988).

[11] P.F. Gonzalez-Daz, Phys. Rev. D54, 6122 (1996).[12] L.D. Landau and E.M. Lifshitz, The Classical Theory of

Fields (Pergamon Press, Oxford, 1975).

[13] J.G. Cramer, R.L. Forward, M.S. Morris, M. Visser, G.Benford and G.A. Landis, Phys. Rev. D51, 3117 (1995).

[14] A. Vilenkin and E.P.S. Shellard, Cosmic Strings andOther Topological Defects (Cambridge Univ. Press, Cam-bridge, 1994).

[15] R. Loudon, The Quantum Theory of Light (Clarendon,Oxford, 1986).

[16] C.M. Urry and P. Padovani, Publ. Ast. Soc. Pacif. 107,803 (1995).

Legend for Figures

Fig. 1. Geometry for gravitational lensing by the neg-ative mass placed at the throat of a ringhole. Off-axislight rays from stellar source S are deflected to detec-tor D by the gravitational repulsion of negative mass M.For embeddible ringholes the two angles denoted wouldbecome slightly different.

Fig. 2. Schematic picture of a galaxy being observedthrough a ringhole tunnel with moving mouths. Thethroat can only be directly accessed if the line of sightlies inside a direct observation cone, < 45. For linesof sight outside that cone, > 45, only the regionsplaced near the ringholes mouths and some secondary

effects induced on the throat could be observed. On thefigure, HIBLR: hight-intensity broad-line region () andLINLR: low-intensity narrow-line region (). The termFriedmann connection, also on this figure, merely de-notes the conventional spacetime that connects the twomouths of the ringhole by going outside the tunnel.

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Pedro F. Gonzalez-Dıaz- Observable effects from spacetime tunneling