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An Analysis of Gravitational Redshift from Rotating Body

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An Analysis of Gravitational Redshift from Rotating Body An analysis of gravitational redshift from rotating body Anuj Kumar Dubey∗ and A K Sen† Department of Physics, Assam University, Silchar-788011, Assam, India. (Dated: July 29, 2021) Gravitational redshift is generally calculated without considering the rotation of a body. Neglect- ing the rotation, the geometry of space time can be described by using the spherically symmetric Schwarzschild geometry. Rotation has great effect on general relativity, which gives new challenges on gravitational redshift. When rotation is taken into consideration spherical symmetry is lost and off diagonal terms appear in the metric. The geometry of space time can be then described by using the solutions of Kerr family. In the present paper we discuss the gravitational redshift for rotating body by using Kerr metric. The numerical calculations has been done under Newtonian approximation of angular momentum. It has been found that the value of gravitational redshift is influenced by the direction of spin of central body and also on the position (latitude) on the central body at which the photon is emitted. The variation of gravitational redshift from equatorial to non - equatorial region has been calculated and its implications are discussed in detail. I. INTRODUCTION from principle of equivalence. Snider in 1972 has mea- sured the redshift of the solar potassium absorption line General relativity is not only relativistic theory of grav- at 7699 A˚ by using an atomic - beam resonance - scat- itation proposed by Einstein, but it is the simplest theory tering technique [5]. Krisher et al. in 1993 had measured that is consistent with experimental data. Predictions of the gravitational redshift of Sun [6]. general relativity have been confirmed in all observations Gravitational redshift has been reported by most of the and experiments. Gravitational redshift of light is one authors without consideration of rotation of a body. Ne- of the predictions of general relativity and also provides glecting the rotation, the geometry of space time can evidence for the validity of the principle of equivalence. be described using the well-known spherically symmetric Any relativistic theory of gravitation consistent with the Schwarzschild’s geometry and information on the ratio M principle of equivalence will predict a redshift. r of a compact object can be obtained from the grav- If, however, we observe on the Earth the spectrum emit- itational redshift, where M and r are mass and radius ′ ted by the atoms located on the Sun, then, its lines ap- respectively. Thus the redshifted angular frequency (ω ) pear to be shifted with respect to the lines of the same and the original angular frequency (ω) of a photon in spectrum emitted on the Earth. Each line with frequency Schwarzschild geometry are related by the relation (page ω will be shifted through the interval ∆ω given by for- 268, of Landau and Lifshitz [1]) ω(V1−V2) mula: ∆ω = c2 where V1 and V2 are the potentials ′ ω ω of the gravitational field at the points of emission and ω = = (1) √gtt 1 rg observation of the spectrum respectively (page 269, of − r Landau and Lifshitz [1]). If we observe on the earth a q 2 spectrum emitted on the sun or the stars, then magni- where rg = (2GM/c ) is the Schwarzschild radius. P tude of V1 is greater than magnitude of V2 and it is clear D Nunez and M Nowakowski in 2010 had obtained an that ∆ω<0, i.e, the shift occurs in the direction of lower expression for gravitational redshift factor of rotating frequency. The phenomenon we have described is called body by using small perturbations to the Schwarzschild’s arXiv:1402.6491v2 [gr-qc] 21 May 2014 the ‘gravitational redshift’. geometry. Two main results of this work include the Adams in 1925 has claimed first about the confirma- derivation of a maximum angular velocity depending tion of the predicted gravitational redshift from the mea- only on the mass of the object and a possible estimates surement of the apparent radial velocity of Sirius B [2]. of the radius [7]. Pound and Rebka in 1959 were the first to experimentally verify the gravitational redshift from nuclear - resonance With this background the present paper is orga- [3]. Pound and Snider in 1965 had performed an im- nized as follows. In Section -II, we derive the expression proved version of the experiment of Pound and Rebka, to for four velocity and frame dragging in kerr field. In measure the effect of gravity, making use of Mossbauer Section -III, we derive the expression for gravitational - Effect [4]. If the Source of radiation is at a height h redshift factor ( ) and redshift (Z) as the pulsar rotates. above an observer then the result found was (0.9990 The derived expressionℜ of gravitational redshift is −15 2gh ± 0.0076) times the value of 4.905 10 c2 predicted applied to calculate the values of gravitational redshift × × for various rotational bodies which includes the Sun and some millisecond pulsars. A pulsar as it rotates, the redshifted values of spectrum shows a periodicity. In ∗ [email protected] Section -IV, we derive the expression for the coefficient † [email protected] of latitude dependence of redshift (κ). We further 2 discuss its variation for different values of latitude at is expressed as which photon is emitted. Finally, some conclusions are 2 2 2 2 2 2 made in Section -V. ds = gttc dt +grrdr +gθθdθ +gφφdφ +2gtφcdtdφ (2) Non-zero components gij of Kerr family are given as fol- II. FRAME DRAGGING IN KERR FIELD lows (page 346, of Landau and Lifshitz [1] and page 261- 263, of Carroll [8]), A. Kerr Field 2 2 ∆ a sin θ rgr gtt = − = (1 ) (3) ρ2 − ρ2 When rotation is taken into consideration spherical symmetry is lost and off - diagonal terms appear in the metric and the most useful form of the solution of Kerr ρ2 grr = (4) family is given in terms of t, r, θ and φ, where t, and r − ∆ are Boyer - Lindquist coordinates running from - to + , θ and φ, are ordinary spherical coordinates in∞ which ∞ 2 φ is periodic with period of 2 π and θ runs from 0 to π. gθθ = ρ (5) − Covariant form of metric tensor with signature (+,-,-,-) 2 2 2 2 2 2 2 2 [(r + a ) a ∆sin θ]sin θ 2 2 rgra sin θ 2 gφφ = − = [r + a + ]sin θ (6) − ρ2 − ρ2 2 2 2 asin θ(rgr e ) π 2 2 rga gtφ = − (7) gφφ(θ = )= (r + a + ) (11) ρ2 2 −π r a r g (θ = ) = ( g ) (12) tφ 2 r with 2 2 2 2 At poles where θ = 0, from equations (8) and (9), if ρ = r + a cos θ (8) we consider charge e = 0, then ρ2 = r2 + a2 and ∆ = 2 2 r rgr + a . − From equations (3), (6) and (7) metric elements gtt, gφφ 2 2 2 ∆= r rg r + a + e (9) and g for Kerr space time at poles become, − tφ J rgr where the parameters e and a(= ) are respectively gtt(θ =0)=1 (13) Mc r2 + a2 charge and rotation parameter of the source. J is the an- − gular momentum of the compact object or central body, which can be also written as J = IΩ. M, I and Ω gφφ(θ =0)=0 (14) are the mass, moment of inertia and angular velocity of the central body respectively. Charge (e) is written as e = Q2 + P 2 , where Q is electric charge and P is gtφ(θ =0)=0 (15) magnetic charge. In case of Kerr solution in 1963 both Q and P vanishesp [9], for Kerr - Newman solution in 1965 only P vanish [10] but in case of Kerr - Newman - Kasuya B. Four - Velocity in Kerr Field solution in 1982 both Q and P are non-vanishing [11]. The values of gtt, gtφ and gφφ for polar and equatorial Four - dimensional velocity (four - velocity) of a par- regions can be obtained from equations (3) to (9). ticle is defined as the four - vector (page 23, of Landau π and Lifshitz [1]) For equatorial plane where θ = 2 , from equations (8) 2 2 and (9), if we consider charge e = 0, then ρ = r and i 2 2 dx ∆= r rgr + a . i − u = (16) From equations (3), (6) and (7) metric elements gtt, gφφ ds and g for Kerr space time at equator becomes, tφ where the index i takes on the values 0,1,2,3 and x0 = ct, x1 = r, x2 = θ, x3 = φ. π rg gtt(θ = )=(1 ) (10) Using equation (2) we can write 2 − r 3 2 2 2 2 2 ds = gttc dt + grrdr + gθθdθ + gφφdφ +2gtφcdtdφ (17) q The components of four - velocity are dxo cdt u0 ut = = (18) 2 2 2 2 2 ≡ ds gttc dt + grrdr + gθθdθ + gφφdφ +2gtφcdtdφ p dx1 dr u1 ur = = (19) 2 2 2 2 2 ≡ ds gttc dt + grrdr + gθθdθ + gφφdφ +2gtφcdtdφ p dx2 dθ u2 uθ = = (20) 2 2 2 2 2 ≡ ds gttc dt + grrdr + gθθdθ + gφφdφ +2gtφcdtdφ p dx3 dφ u3 uφ = = (21) 2 2 2 2 2 ≡ ds gttc dt + grrdr + gθθdθ + gφφdφ +2gtφcdtdφ From equations (18) and (21) we can writep u3 uφ dφ = (22) u0 ≡ ut cdt For a sphere, the photon is emitted at a location on its From the Lagrangian of the test particle, we can obtain surface where dr = dθ = 0, when the sphere rotates. the momentum of the test particle as, So the expressions for four - velocity becomes, ∂£ P = = g x˙ j (29) 1 i i ij u0 ut = (23) ∂x˙ ≡ dφ 2 dφ gtt + gφφ( cdt ) +2gtφ( cdt ) Thus corresponding momentum in the coordinates of t, q r, θ, and φ are given by 1 r 2 u u = 0 (24) Pt E = c gttt˙ + cgtφφ˙ (30) ≡ ≡− 2 θ u u = 0 (25) Pr = grrr˙ (31) ≡ dφ P = g θ˙ (32) u3 uφ = ut (26) θ θθ ≡ cdt Hence four - velocity of an object in Kerr field can be Pφ L = cgtφt˙ + gφφφ˙ (33) expressed as, ≡ Since the space time of the Kerr family is stationary and dφ axially symmetric, the momenta P and P are conserved ui = (ut, 0, 0,ut ) (27) t φ cdt along the geodesics.
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