Petkov - Elevator.pdf - Energy from the Vacuum - andreas50_54

ThreelightraysareemittedsimultaneouslyinanelevatoratrestintheEarth’sgravitationalﬁeld(representing

anon-inertialreferenceframeN g )frompointsD,R,andStowardpointM.LetIbeareferenceframeinitially

atrestwithrespecttoN g whichstartstofallinthegravitationalﬁeldatthemomentthelightraysareemitted.

TheemissionoftheraysissimultaneousinN g aswellasinI.AtthenextmomentanobserverinIseesthatthe

elevatormovesupwardwithanaccelerationg.Thereforethethreelightraysarrivesimultaneouslynotatpoint

M,butatOsinceforthetimet=r/ctheelevatormovesatadistance=gt 2 /2=gr 2 /2c 2 .Asthesimultaneous

arrivalofthethreeraysatthepointOinIisanabsoluteevent(thesameinallreferenceframes)beingapoint

event,itfollowsthattheraysarrivesimultaneouslyatOasseenfromN g aswell.Sinceforthesamecoordinate

timet=r/cinN g thethreelightraystraveldierentdistancesDOr,SO=r+,andRO=r−before

arrivingsimultaneouslyatpointOanobserverintheelevatorconcludesthattheaveragedownwardvelocity¯c #

ofthelightraypropagatingfromStoOisslightlygreaterthanc

¯c # = r+

t c 1+ gr

2c 2

Theaverageupwardvelocity¯c " ofthelightraypropagatingfromRtoOisslightlysmallerthanc

¯c " = r−

t c 1− gr

2c 2

ThevectorformoftheaveragelightvelocityinN g canbeobtainedifR,S,M,andOaretakentolieonaline

makingananglewithg:

¯c g =c

1+ g·r

2c 2

(1)

—

S r

= 1 2 gt 2 =gr 2 /2c 2

O ?

—

Figure1.ThreelightrayspropagateinanelevatoratrestintheEarth’sgravitationalﬁeld.Afterhaving

beenemittedsimultaneouslyfrompointsD,R,andStheraysmeetatO(theraypropagatingfromD

towardM,butarrivingatO,representstheoriginalthoughtexperimentconsideredbyEinstein).Thelight

raysemittedfromRandSareintroducedinordertodeterminetheexpressionfortheaverageanisotropic

velocityoflightinagravitationalﬁeld.Ittakesthesamecoordinatetimet=r/cfortheraystotravel

thedistancesDOr,SO=r+,andRO=r−.Thereforetheaveragevelocityofthedownward

rayfromStoOis¯c # =(r+)/tc(1+gr/2c 2 );theaveragevelocityoftheupwardrayfromRtoOis

¯c " =(r−)/tc(1−gr/2c 2 ).

Petkov - Elevator.pdf