“TheGravitationalRed-Shift” R.F.EvansandJ.Dunning-Davies, DepartmentofPhysics, UniversityofHull, HullHU67RX, England. [email protected] Abstract. Attentionisdrawntothefactthatthewell-knownexpressionforthered-shiftof spectral lines due to a gravitational field may be derived with no recourse to the theory of general relativity. This raises grave doubts over the inclusion of the measurementofthisgravitationalred-shiftinthelistof crucial testsofthetheoryof generalrelativity. Introduction. In most of the standard texts concerned with the General Theory of Relativity [1,2,3], reference is made to three crucial tests of the theory; - the well-known Advance of the Perihelion of the planet Mercury, the Gravitational Deflection of Light Rays, and the Gravitational Shift of Spectral Lines. Weinberg [3] does draw attentiontotwootherpossibletests,butstillretainstheabovethreeinhislist. Thesituationseemsquiteinterestingasfarasthethirdoftheseapparently crucial tests is concerned. In the first of the above references, attention seems to be concentratedsolelyontheeffectbeingduetogeneralrelativisticeffects.Inthethird reference, it is noted, quite clearly, that the effect arises due to the principle of equivalencealone;theEinsteinFieldEquationsarenotconcernedinthederivationof theformulaatall.Itisonlyinthesecondreferencelistedherethatattentionisdrawn tothefactthat,aswellasthetestonlybeingoneofthevalidityoftheprincipleof equivalence,alternativederivationsoftheresultexistalso.Itispossiblynotsurprising, therefore,thatfewpeopleseemtorealisethatthesaidresultmaybederivedwithno recourse to the general theory of relativity whatsoever, nor to the principle of equivalence. It is possibly surprising, therefore, to find that one of the alternative derivationsmentionedinthesecondreferenceisthisoneinwhichthegeneraltheory ofrelativityplaysnopart.Theendresultisthattheexistenceofthisderivationisnot widelyknownandleadstosituations,suchasthatreportedhere,inwhichtheresultis derivedafreshwithnoknowledgeoftheearlierwork. TheGravitationalRed-Shift. Conservation of energy yields the fact that the sum of the kinetic and potential energies is a constant. If a particle of mass m is moving under the influence of a gravitationalfieldgeneratedbyamassivecentralbodyofmass M,Newton’slawof gravitationshowsthatthepotentialenergyisgivenby-GMm /r,where GisNewton’s universalconstantofgravitationand risthedistanceoftheparticlefromthecentral massivebody. However,whatofthekineticenergywhich,foranormalmaterialparticle,istaken to be one half the product of the particle’s mass with the square of its velocity? Obviously,suchaformulawouldnotapplyinthecaseofa‘particle’oflight,which haszeromass.However,thekineticenergyof aphotonisgivenbyhν,where his Planck’sconstantand νisthefrequencyofthephoton.Ifthemass-energyrelation E= mc 2, whichrelatesthekineticenergytotheproductofmassandthesquareofthespeedof light,isintroduced,thenan‘effectivemass’for thephotonmaybededuced andis givenby m= hν/c2. Theequationexpressingconservationofenergythenbecomes hν- GMm /r= hν- GMh ν/rc 2=constant. Thisequationimmediatelyallowsthewellestablishedexpressionforthegravitational red-shift to be deduced. For example, if as r → ∞, ν → ν∞, the equation of conservationofenergybecomes 2 hν- GMh ν/rc = hν∞ or ν −ν GM ∞ = − , ν rc 2 whichisthedesiredresult. Discussion. Intheabovederivationoftheexpressionforthegravitationalred-shift,noappeal hasbeenmadetoanyaspectofthetheoryofgeneralrelativity,noteventheprinciple of equivalence. Hence, the question must be raised as to how and why the measurement of the gravitational red-shift could ever be considered a real test of generalrelativity,letalonea crucial testasissooftenclaimed?Itseemssurprising that,oncetheabovedeductionofan‘effectivemass’forthephotonwasrecognised, thissimplederivationofthered-shiftformuladidnotbecomemorewidelyrecognised. It is possibly even more surprising to note that, although the writing-up is slightly different,thisderivationisincludedinthesecondreferencelisted,but,eventhere,itis stillregardedasbeingacrucialtestofgeneralrelativity.Intruth,itwouldseemthat theresultforthered-shiftofspectrallinesduetotheactionofagravitationalfieldhas nothing specifically to do with the theory of general relativity. It is a result which drawsonmoremodernresultsduetosuchasPlanckandPoincaré,but, apart from those,isdeducedfromnotionsofNewtonianmechanicsalone.Assuch,itseemsto have no place in a list of crucial tests of general relativity, although the theory of generalrelativityobviouslymustnotcontradictthisresult.Thispointhasbeenmade alsobyLavenda[4]inanarticlewherehealsoshowsthat,ofthethreeusuallylisted crucial tests, the deflection of light and the perihelion advance can be treated as diffractionphenomenaonthebasisofFermat’sprincipleandthemodificationofthe phaseofaBesselfunctionintheshort-wavelengthlimit.Incidentally,Lavendaalso dealswiththeproblemofthetimedelayinradarechoesbythismeans.Wheredoes allthisleavetheGeneralTheoryofRelativity? References. [1]Tolman,R.C.,“Relativity,ThermodynamicsandCosmology”, OxfordUniversityPress,Oxford,1962. [2]Adler,R.,Bazin,M.&Schiffer,M.,“IntroductiontoGeneralRelativity“, McGraw-Hill,NewYork,1965. [3]Weinberg,S.,“GravitationandCosmology”, J.Wiley&Sons,NewYork,1972. [4]Lavenda,B.H.,“ThreeTestsofGeneralRelativityviaFermat’sPrincipleand thePhaseofBesselFunctions”,arXiv:math-ph/0310054.