Physics Letters BVol. 687, Nos. 2-3 (2010) pp. 110113DOI: 10.1016/j.physletb.2010.03.029c Elsevier B. V.RADIALMOTIONINTOANEINSTEIN-ROSENBRIDGENikodemJ. PoplawskiDepartment of Physics, IndianaUniversity, SwainHall West,727 East Third Street, Bloomington, IN47405, USAWeconsidertheradial geodesicmotionofamassiveparticleintoablackholeinisotropiccoor-dinates,whichrepresentstheexteriorregionofanEinstein-Rosenbridge(wormhole). Theparticleenters the interior region,whichis regular andphysically equivalenttothe asymptotically atexte-rior of a white hole, and the particles proper time extends to innity. Since the radial motion into awormhole after passing the event horizon is physically dierent from the motion into a Schwarzschildblackhole,Einstein-RosenandSchwarzschildblackholesaredierent,physicalrealizationsofgen-eral relativity. Yet for distant observers, both solutions are indistinguishable. We show that timelikegeodesicsintheeldofawormholearecompletebecausetheexpansionscalarintheRaychaudhuriequation has a discontinuity at the horizon, and because the Einstein-Rosen bridge is represented bythe Kruskal diagram with Rindlers elliptic identication of the two antipodal future event horizons.TheseresultssuggestthatobservedastrophysicalblackholesmaybeEinstein-Rosenbridges,eachwithanewuniverseinsidethatformedsimultaneouslywiththeblackhole. Accordingly, ourownUniversemaybetheinteriorofablackholeexistinginsideanotheruniverse.Keywords: Blackhole,Isotropiccoordinates,Wormhole.I. ISOTROPICCOORDINATESThe interval of the static, spherically symmetric, gravitational eld in vacuum, expressed in isotropic coordinates,was found by Weyl [1]:ds2=(1 rg/(4r))2(1 + rg/(4r))2c2dt2(1 + rg/(4r))4(dr2+ r2d2), (1)where 0 r < is the radial coordinate, d is the element of the solid angle, and rg = 2GM/c2is the Schwarzschildradius. This metric does not change its form under the coordinate transformation:r r

=r2g16r, (2)and is Galilean forr . Therefore it is also Galilean forr 0, describing an Einstein-Rosen bridge (wormhole):twoSchwarzschildsolutionstotheEinsteineldequations(ablackholeandwhitehole)connectedatthesingular(det g= 0)surfacer=rg/4(commoneventhorizon)[2,3]. ThenonzerocomponentsoftheRiemanncurvaturetensor for this metric are given byR00 = R00 = Rrr = Rrr =rg2r3(1 + rg/(4r))6,R0r0r = R = rgr3(1 + rg/(4r))6, (3)sotheKretschmannscalarisniteeverywhere: RR=12r2gr6(1 + rg/(4r))12, goingtozeroasr andr 0. The Einstein-Rosen metric forr>rg/4 describes the exterior sheet of a Schwarzschild black hole (theElectronicaddress: nipoplaw@indiana.edu2transformation of the radial coordinate r rS = r(1+rg/(4r))2brings the interval (1) into the standard Schwarzschildform [4]). The spacetime given by the metric (1) forr< rg/4 is regarded by an observer atr> rg/4 as the interiorofablackhole. Becauseoftheinvarianceofthemetric(1)underthetransformation(2),thisinteriorisanimageof the other exterior sheet. This situation is analogous to the method of image charges for spheres in electrostatics,where the interaction between an electric charge situated at a distancerfrom the center of a conducting sphere ofradiusR