Gravitational Shift for Beginners

Transcription

1 Gavitational Shift fo Beginnes This pape, which I wote in 26, fomulates the equations fo gavitational shifts fom the elativistic famewok of special elativity. Fist I deive the fomulas fo the gavitational edshift and then the fomulas fo the gavitational blueshift. by. A. Fino 26 Keywods: edshift, blueshift, gavitational shift, gavitational edshift, gavitational blueshift, total elativistic enegy, gavitational potential enegy, elativistic kinetic enegy, consevation of enegy, wavelength, fequency, special theoy of elativity, geneal theoy of elativity. Contents. Gavitational edshift 2. Gavitational Blueshift 3. Summay of Fomulas Appendix : Nomenclatue Gavitational Shift fo Beginnes- v. Copyight odolfo A. Fino. All ights eseved.

2 . Gavitational edshift Let's conside a sta of mass M and adius that emits a photon as shown in Fig. The photon tavels though empty space an abitay distance befoe eaching ou planet. We shall also conside an obseve located on Eath who measues the fequency f f) of the eceived photon. Fig : Gavitational edshift. We want to calculate the wavelength shift poduce by the sta's gavity on the emitted photon. Thus, if the initial fequency of the emitted photon was f on the suface of the sta), the final fequency of the photon, just befoe detection, will be f ). Thus, we want to find the fomula that gives the fequency, f, as a function of and f. Accoding to the law of consevation of enegy we can wite Consevation of enegy E)+U ) E + U.) Because photons ae consideed to be massless, thei total enegy, E, is identical to thei kinetic enegy, K. Gavitational Shift fo Beginnes- v. Copyight odolfo A. Fino. All ights eseved. 2

3 E) Total o kinetic) enegy of the photon at a distance fom the cente of the sta E Total o kinetic) enegy of the photon on the suface of the sta U ) Gavitational potential enegy of the photon at a distance fom the cente of the sta U Gavitational potential enegy of the photon on the suface of the sta The enegy of the photon on the suface of the Eath can be expessed as E)h f ).2) The enegy of the photon on the suface of the sta can be expessed as E hf.3) The Gavitational potential enegy is given by Gavitational potential enegy U ) G M m.4) Whee m is the equivalent mass of the photon and can be deived fom the following equation hf m.5) The fist side of this equation is the enegy of the photon accoding to Planck's theoy of electomagnetic adiation. The second side of the equation is the same enegy accoding to Einstein's fomula of equivalence of mass and enegy. Hence, solving this equation fo m yields Equivalent mass of the photon m h f.6) eplacing the vaiable m in equation 3.4), by the value found in eq. 3.6) we have U ) G M h f.7) This is the potential enegy of the photon at a distance fom the cente of the sta. Similaly, the potential enegy of the photon on the suface of the sta is U G M h f.8) Fom the equation of consevation of enegy,.), and fom equations.2),.3),.7) and.8) we can wite Gavitational Shift fo Beginnes- v. Copyight odolfo A. Fino. All ights eseved. 3

4 hf ) G M h f ) hf G M h f.9) Dividing by h both sides and taking a common facto, f ), on the fist side of the equation and a common facto, f, on the second side, we have f ) G M c ) f 2 G M c ) 2.) Solving this equation fo f ) yields G M ) f ) G M f.) This fomula gives the fequency of the photon as a function of the distance fom the cente of the sta. This is the fequency an obseve would measue if gavitational shifts wee the only shifts acting in the univese of couse this is not tue). Now if the distance is infinite the pevious fomula tansfoms into f ) f G M c ) f 2.2) Now we define the special Schwazchild adius as G M.3) It is wothwhile to obseve that, because this appoach doesn't use geneal elativity, the value of the special Schwazchild adius is half of what we get when we apply Einstein's field equations. Inseting this value, the two pevious equations become and f ) ) f.4) f ) f.5) Now I shall expess equations.4) and.5) in tems of the wavelength of the photon. To achieve that we conside the following fomula fo the speed of light Gavitational Shift fo Beginnes- v. Copyight odolfo A. Fino. All ights eseved. 4

5 c λ ) T ) λ )f ).6) Thus f ) c λ ).7) Now I use eq..7) to eliminate f fom eq..4) and I solve it fo λ ) λ ) Doing a simila wok, eq..5) becomes Now, we assume that Then I divide by ) λ ) λ.8) ) λ.9) < this is, the sta is not a black hole. Because > ) >.2) ) both sides of inequation.2). This gives )>.2) Compaing eq..8) with inequation.2) we see that the quantity inside the paenthesis is the same in both expessions, theefoe λ ) must be geate than λ Mathematically λ ) > λ.22) Because the obseved wavelength of the photon measued by an obseve) is geate than its initial wavelength when it was emitted by the sta), the enegy of the eceived photon is less that the enegy of the emitted photon. Theefoe we say that thee is a gavitational ed shift. Gavitational Shift fo Beginnes- v. Copyight odolfo A. Fino. All ights eseved. 5

6 2. Gavitational Blueshift Because the deivation of the fomulas fo both shifts is identical with the diffeence, in the end, that we solve the equations fo diffeent paametes), I shall use a slightly diffeent method to make it moe inteesting. Specifically I shall use a distance H which is the distance fom the suface of the sta to a distant photon that tavels towads the sta. In the end this photon is swallowed by the sta. I assume that thee is an obseve on the suface of the sta who measues I don't know how) the fequency and the wavelength of the photon at its aival. Fig 2: Gavitational blueshift. Accoding to the law of consevation of enegy we can wite Consevation of enegy E H + U H E + U 2.) Because photons ae consideed to be massless, thei total enegy E is identical to thei kinetic enegy K. Gavitational Shift fo Beginnes- v. Copyight odolfo A. Fino. All ights eseved. 6

7 E H Total o kinetic) enegy of the photon at a distance H fom the suface of the sta E Total o kinetic) enegy of the photon on the suface of the sta U H Gavitational potential enegy of the photon at a distance H fom the suface of the sta U Gavitational potential enegy of the photon on the suface of the sta The photon enegy at position can be expessed as The photon enegy at position 2 can be expessed as The Gavitational potential enegy is given by E H hf H 2.2) E hf 2.3) Gavitational potential enegy U G M m 2.4) Whee m is the equivalent mass of the photon and can be deived fom the following equation hf m 2.5) The fist side of this equation is the enegy of the photon accoding to Planck's theoy of electomagnetic adiation. The second side of the equation is the same enegy accoding to Einstein's fomula of equivalence of mass and enegy. Hence, solving this equation fo m yields Equivalent mass of the photon m h f 2.6) eplacing the vaiable m in equation 2.4), by the value found in eq. 2.6) we have Gavitational potential enegy U G M h f 2.7) Thus, the potential enegy of the photon at a distance H fom the suface of the sta is U H G M h f H + H 2.8) Similaly, the potential enegy of the photon on the suface of the sta is Gavitational Shift fo Beginnes- v. Copyight odolfo A. Fino. All ights eseved. 7

8 U G M h f 2.9) Fom the equation of consevation of enegy, 2.), and fom equations 2.2), 2.3), 2.8) and 2.9) we can wite h f H G M h f H + H h f G M h f 2.) Dividing by h both sides and taking a common facto f H facto f on the second one, we have f H G M + H ) f G M c ) 2 on the fist side and a common 2.) Solving this equation fo f yields G M f + H ) f H 2.2) G M This fomula gives the fequency of the photon on the suface of the sta as a function of its fequency at a distance H. Now we conside that the distance H is infinite. Thus we can take the limit of the pevious expession when H tends to infinity. Mathematically ) G M + H lim f lim H H G M H) f 2.3) The esult of this limit is f G M f 2.4) Whee f is the fequency of the photon at an infinite distance fom the sta, in othe wods: f H ) f. Now we define G M 2.5) It is wothwhile to obseve that, because this appoach doesn't use geneal elativity, the value of this adius is half of what we get when we apply Einstein's field equations. Gavitational Shift fo Beginnes- v. Copyight odolfo A. Fino. All ights eseved. 8

9 Combining equations 2.4) and 2.5) yields f f 2.6) Now I shall expess equations 2.4) and 2.6) in tems of the wavelength of the photon. To achieve that we conside the following fomula c λ T λ f 2.7) Combining equations 2.4) and 2.5) and solving fo λ we get λ Combining equations 2.6) and 2.5) yields G M c ) λ 2 2.8) λ ) λ 2.9) Let's intepet this esult. If > then the quotient / is less than. Mathematically This means that that the following inequation must be tue < 2.2) ) < 2.2) Thus, in accodance to eq. 2.6) we have the following inequation λ < λ 2.22) Whee λ is the wavelength of the photon when H is infinite. This is λ H ) λ. Theefoe the wavelength of the photon deceases as it gets close to the sta. But because the enegy of the photon is popotional to the invese of its wavelength, as the following fomula shows, E hc λ 2.23) the enegy of the photon inceases as its wavelength deceases. In othe wods, the enegy of the photon inceases as it gets close to the sta. This effect is known as blue shift. Gavitational Shift fo Beginnes- v. Copyight odolfo A. Fino. All ights eseved. 9

10 Fom the point of view of the two types of enegy involved in the pocess: potential and kinetic o total), we can say that, as the photon appoaches the sta, it loses gavitational potential enegy and gains kinetic enegy which is equal to its total elativistic enegy). Thus, the enegy of the photon on the suface of the sta is geate than its enegy at infinity. Mathematically we expess this fact as follows Suface E hc λ ) Infinity > E hc λ ) 2.24) 3. Summay of Fomulas The following tables summaize the final fomulas deived in the pevious sections. ed shift G M Fequency measued on Eath Wavelength measued on Eath Table : eshift equations Fomulas when is finite) f ) λ ) ) f ) λ Fomulas when is infinite) f λ ) f ) λ Blue shift G M Fequency on the suface of the sta Wavelength on the suface of the sta Fomulas when is finite) f ) ) f ) λ ) ) λ ) Fomulas when is infinite) f λ f ) λ Gavitational Shift fo Beginnes- v. Copyight odolfo A. Fino. All ights eseved.

11 Table 2: Blueshift equations Appendix Nomenclatue I shall use the following nomenclatue fo the constants and vaiables used in this pape c speed of light in vacuum h Planck's constant G Gavitational constant also known as constant of gavitation, constant of gavity, gavitational foce constant, univesal constant of gavity, univesal gavitational constant, Newton's gavitational constant, Newtonian gavitational constant, etc.) distance fom the cente of the sta to an obseve on Eath. Also distance fom a distant photon to the cente of the sta H distance fom the suface of the sta to a distant photon that tavels towads the sta used in the deivation of blueshifts only) m equivalent mass of the photon M mass of the sta adius of the sta special Swazchild adius of the sta special elativity's fomula) I denoted this quantity this way to diffeentiate it fom the coect Schwazchild adius, deived fom geneal elativity, which is geneally denoted by S E total elativistic enegy of the photon K kinetic elativistic enegy of the photon E) total o kinetic) enegy of the photon at a distance fom the cente of the sta o, equivalently, the enegy of the photon on the suface of the Eath E total o kinetic) enegy of the photon on the suface of the sta U ) gavitational potential enegy of the photon at a distance fom the cente of the sta, o, equivalently, the gavitational potential enegy of the photon on the suface of the Eath U gavitational potential enegy of the photon on the suface of the sta f fequency of the photon on the suface of the sta f fequency of the photon at an infinite distance fom the sta λ wavelength of the photon on the suface of the sta λ wavelength of the photon at an infinite distance fom the sta f ) fequency of the photon at a distance fom the cente of the sta o, equivalently, the fequency of the photon on the suface of the Eath T peiod I use K fo kinetic enegy) T ) peiod of the photon at a distance fom the cente of the sta λ ) wavelength of the photon at a distance fom the cente of the sta o, equivalently, the wavelength of the photon on the suface of the Eath Gavitational Shift fo Beginnes- v. Copyight odolfo A. Fino. All ights eseved.