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Assam, = 2 steShazcidrdu.P radius. Schwarzschild the is ) √ ω g tt = q 1 ω − r r ω g κ fapoo in photon a of ) .W further We ). ω (1) ′ ) 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 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 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 . So we obtain two constants of mo- tion: one is corresponding to the conservation of (E) and the other is the angular momentum (L) about C. Frame Dragging in Kerr Field the symmetry axis. we can write from equations (30) and (33) The Lagrangian £ of the test particle (with mass m) can be expressed as ˙ gφφE + cgtφL ct = 2 (34) cgtφ cgttgφφ g x˙ ix˙ j − £ = ij (28) 2 and (Dot over a symbol denotes ordinary differentiation with ˙ gtφE + cgttL φ = 2 (35) respect to an affine parameter ξ) −cg cgttgφφ tφ − 4

dφ Using equations (34) and (35), we can obtain cdt rotating . Due to the influence of gravity frame dragging or dragging of inertial frame arises in the Kerr ˙ φ dφ φ u dφ gtφE + cgttL metric. cdt is termed as angular velocity of frame drag- = t = = (36) ct˙ u cdt −gφφE + cgtφL ging [12]. The equation (37) is the general expression of dφ angular velocity of frame dragging ( cdt ) in Kerr field. Substituting the value of gtt, gφφ, and gtφ from equations Following (Cunningham and Bardeen 1972 [13]), the pho- (3), (6) and (7) in above equation (36), ton trajectory is independent of its energy and may be L 2 described by the parameter E , which is related to the asin θrgr rg r direction cosines of a beam of radiation with respect to dφ ρ2 E + c(1 ρ2 )L = − the φ - direction and θ - direction in the local rest frame. 2 2 2 cdt − 2 2 rgra sin θ 2 asin θrgr 2 In case of photon the energy (E) and linear momentum (r + a + 2 )sin θE + c ρ L − ρ (p) are expressed as

2 2 2 2 4 cL rgr rg ra E = p c + m0c (41) dφ sin2θ (1 ρ2 )+ ρ2 E = − 2 (37) where m0 is rest mass of the photon which is zero. So crgra 2 2 rgra 2 cdt 2 L + E(r + a + 2 sin θ) above equation becomes − ρ ρ Alternatively following Landau and Lifshitz ([1] page E = pc (42) dφ 351), we can also find cdt in the following way: The four momentum of the test particle (with mass m) is The angular momentum (L) about the symmetry axis may be expressed as dxi pi = m (38) ds L = pb (43)

£ 2 We may note that the symbol ‘ ’ stands for Lagrangian cdt rgra E 2 2 rg ra 2 and ‘L’ for angular momentum about the symmetry axis. m = 2 cL + (r + a + 2 sin θ) (39) ds − ρ ∆ ∆ ρ It is assumed that b≫ rg and b ≫ a. Here b is per- pendicular between the axis of rotation of the central body and direction of ray of light. dφ cL rgr rgra m = (1 )+ E (40) Considering the ray of light which is emitted radially out- ds ∆sin2θ ρ2 ρ2∆ − ward from the surface of the (Pulsar or Sun) with radius ‘R’, we can substitute, b = Rsinφsinθ in above From above equations (39) and (40), we can find dφ as cdt equation (43) and finally we can write

cL rg r rg ra 2 (1 2 )+ 2 E dφ sin θ ρ ρ L = pRsinφsinθ (44) = − 2 cdt rgra 2 2 rgra 2 ρ2 cL + E(r + a + ρ2 sin θ) − where φ =Ωt, is measured from the azimuthal axis The above expression is identical to the expression (37) , which is defined as the direction when source of light obtained earlier, where E and L are conserved energy and (on the spherical surface) is nearest to the observer. 2π components of angular momentum along the axis of sym- Ω= T , where T is period of rotation of star (Pulsar or metry of the field respectively. Sun). Frame dragging is a general relativistic feature of all so- Substituting the value of E and L from equations (42) dφ lutions to the Einstein field equations associated with and (44) in the expression (37) of cdt ,

pcRsinφsinθ rg r rgra dφ sin2 θ (1 ρ2 )+ ρ2 (pc) = − 2 (45) rgra 2 2 rg ra 2 cdt 2 (pcRsinφsinθ) + (pc)(r + a + 2 sin θ) − ρ ρ

Rsinφsinθ rg r rg ra dφ sin2θ (1 ρ2 )+ ρ2 (φ, θ)= − 2 (46) rgra 2 2 rg ra 2 cdt 2 (Rsinφsinθ) + (r + a + 2 sin θ) − ρ ρ

π 2 2 For equatorial plane where θ = 2 and ρ = r , the above The above expression (47) is the expression of frame drag- dφ dφ expression (46) of cdt can be written as: ging ( cdt ) on the equatorial plane.

rg rga dφ π (1 r )(Rsinφ)+ r (φ, θ = )= − 2 (47) cdt 2 rga (Rsinφ) + (r2 + a2 + rga ) − r r 5

Again for any general θ and at φ = π , the expression for ∂Ψ ∂Ψ 2 k1 = kx = = (57) dφ −∂x1 − ∂x cdt from equation (46) can be written as:

R rgr rg ra dφ π sinθ (1 ρ2 )+ ρ2 ∂Ψ ∂Ψ (φ = ,θ)= − 2 k2 = ky = = (58) rgra 2 2 rg ra 2 2 cdt 2 2 (Rsinθ) + (r + a + 2 sin θ) −∂x − ∂y − ρ ρ (48) π 2 2 Substituting θ = 2 and ρ = r , the above expression ∂Ψ ∂Ψ dφ k3 = kz = 3 = (59) (48) of cdt can be written as: −∂x − ∂z rg rga Hence we can write the - four vector dφ π π R(1 r )+ r (φ = ,θ = )= − 2 (49) rga rg a ω cdt 2 2 R + (r2 + a2 + ) k = (k , k , k , k ) = ( , k , k , k ) (60) − r r i 0 1 2 3 c x y z k (= ω ) is the time component of four - wave vector and III. FROM 0 c this frequency is measured in terms of the world time. ROTATING BODY Frequency measured in terms of (τ) is defined as A. Frequency in Kerr Field ′ ∂Ψ ω = (61) Let f be any quantity describing the field of the wave. − ∂τ For a plane monochromatic wave f has the form (page ′ This frequency ω is different at different point of space. 140, of Landau and Lifshitz [1]) 0 0 i ′ ∂Ψ ∂x ∂x f = aej(k.r−ωt+α) = aej(kix +α) (50) ω = = ω = ωu0 (62) −∂x0 ∂τ c∂τ

0 where j = √ 1, and i used as superscript and subscript ∂x 0 with indices− i=0,1,2,3. as c∂τ is nothing but u . Substituting the value of u0 in above equation (62) from k is propagation constant and ki is the wave four - vector. We write the expression for the field in the form equation (23) ′ ω f = aejΨ (51) ω = (63) dφ 2 dφ i gtt + gφφ( cdt ) +2gtφ( cdt ) where Ψ kix + α, is defined as eikonal. ≡− Over small region of space and time intervals the eikonal ′ q Ψ can be expanded in series to terms of first order, we This ω is the frequency measured by a distant observer have in terms of proper time (τ) and ω is the frequency measured in terms of the world time (t). ∂Ψ ∂Ψ Ψ=Ψ + r. + t (52) 0 ∂r ∂t Alternative method: Frequency in Kerr field

As a result one can write (page 141, of Landau and Lif- ′ Any observer measures the frequency (ω ) of a photon shitz [1]) following the null xi(ξ) which can be calculated ∂Ψ by the expression given as (page 217 of Carroll 2004 [8] ki = (53) −∂xi and page 108 of Straumann 1984 [14], j where ki is the wave four - vector and the components of ′ dx dx ω = ui i = uig (64) the four - wave vector ki are related by dξ ij dξ i kik = 0 (54) ′ t 2 ˙ ˙ φ ˙ ˙ or ω = u (gttc t + cgtφφ)+ u (gtφct + gφφφ) (65) ∂Ψ ∂Ψ Using equation (30) and (33) in the above equation (65), i = 0 (55) ∂x ∂xi we can write ′ This equation called the eikonal equation is fundamental ω = ut( E)+ uφ(L) (66) equation of geometrical optics. − Using equation (53), the components of wave four - vector Again using equation (26) in above equation (66) we can are write:

∂Ψ ∂Ψ ω ′ dφ k = = = (56) ω = ut( E + L) (67) 0 −∂x0 −∂ct c − cdt 6 or B. Gravitational redshift factor and redshift

In , redshift (Z) and redshift factor ( ) are defined as [15,16], ′ ℜ 0 ω = ωu 1 ω = = ′ (68) Z +1 ℜ ω A redshift of zero corresponds to an un-shifted line, whereas Z < 0 indicates blue-shifted emission and Z > 0 The above expression is identical to the expression (62) -shifted emission. A redshift factor of unity corre- dφ obtained earlier. Here we define ( E + cdt L) as ω, where sponds to an un-shifted line, whereas < 1 indicates ω can be identified as the frequency− corresponding to red-shifted emission and > 1 -shiftedℜ emission. world time as in equation (56) and equation (62). From equations (63) andℜ (68), we can write redshift fac- As a result, we can write the expression of frequency mea- tor ( ) as: sured in terms of proper time or as observed by distant ℜ observer as: dφ 2 dφ (φ, θ)= gtt + gφφ( ) +2gtφ( ) (69) ℜ r cdt cdt Using equation (68), we can obtain redshift (Z) from ′ ω ω = above equation (69). dφ 2 dφ If we ignore the electric and magnetic charges contribu- gtt + gφφ( cdt ) +2gtφ( cdt ) q tion, then using the values of metric coefficients gtt, gφφ, and gtφ from equations (3), (6) and (7) of Kerr family dφ in equation (69) and substituting the value of cdt from equation (46), we can obtain the redshift factor ( ) and The above expression is identical to the expression (63) corresponding redshift (Z) for the values of θ fromℜ zero π obtained earlier. Thus the two approaches give us identi- (pole) to 2 (equator) and φ from zero to 2π. cal expression for frequency observed by distant observer.

r r 2(φ, θ)=(1 g ) ℜ − r2 + a2cos2θ

2 2 Rsinφsinθ rgr rg ra 2 2 rgra sin θ 2 sin2θ (1 r2+a2cos2θ )+ r2+a2cos2θ 2 (r + a + )(sin θ)( − 2 ) 2 2 2 rg ra 2 2 rg ra 2 − r + a cos θ 2 2 2 (Rsinφsinθ) + (r + a + 2 2 2 sin θ) − r +a cos θ r +a cos θ

2 Rsinφsinθ rgr rg ra asin θ(rg r) 2 sin2θ (1 r2+a2cos2θ )+ r2+a2cos2θ +2 (sin θ)( − 2 ) (70) 2 2 2 rg ra 2 2 rg ra 2 r + a cos θ 2 2 2 (Rsinφsinθ) + (r + a + 2 2 2 sin θ) − r +a cos θ r +a cos θ

We can replace ‘r’ by ‘R’ which is the radius of the star. (Ω) to be zero (or spin parameter, a=0), then we can Thus the redshift clearly depends on the co-ordinates θ obtain the corresponding gravitational redshift from a and φ. static body of same mass (Schwarzschild Mass). If the observer is on the axis of rotating star, we sub- π stitute θ = 0 and get the expression for redshift factor Now at θ = 2 for equatorial plane, substituting as: the values of gtt, gφφ, and gtφ from equations (10), (11) and (12) and dφ (φ, θ = π ) from equation (47) in r r cdt 2 (φ, θ =0)= (1 g ) (71) expression (69) we get: ℜ − r2 + a2 r When we consider the rotation velocity of central body 7

1.8 1.8 1.6 PSR J 1748-2446ad 1.6 PSR B 1937+21 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6

0.6 Redshift (Z) 0.4 Redshift (Z) 0.4 0.2 0.2 0

0 0 45 90 135 180 225 270 315 360 0 45 90 φ 135 (in degree) 180 225 270 315 360 φ (in degree)

2.2 3 2 1.8 PSR J 1909-3744 2.5 PSR 1855+09 1.6 1.4 2 1.2 1 1.5 0.8 1 0.6 Redshift (Z) 0.4 Redshift (Z) 0.5 0.2 0 0 0 0 45 90 45 90 135 180 225 270 315 360 135 180 225 270 315 360 φ (in degree) φ (in degree)

FIG. 1. Shows the variation of redshift (Z) versus φ for pulsars 1-4 (listed in TABLE - I) from φ = 0 to 360o, at fixed θ = 90o. 8

5 6 4.5 PSR J 0737-3039 A 5 PSR 0531+21 4 3.5 4 3 3 2.5 2 2

1.5 Redshift (Z) Redshift (Z) 1 1 0.5 0

0 0 45 90 135 180 225 270 315 360 0 45 90 φ 135 (in degree) 180 225 270 315 360 φ (in degree)

6 1.1 5 PSR B 1534+12 1 PSR B 1913+16 0.9 4 0.8 3 0.7 2 0.6

Redshift (Z) Redshift (Z) 0.5 1 0.4 0 0.3 0 0 45 90 45 90 135 180 225 270 315 360 135 180 225 270 315 360 φ (in degree) φ (in degree)

FIG. 2. Shows the variation of redshift (Z) versus φ for pulsars 5-8 (listed in TABLE - I) from φ = 0 to 360o, at fixed θ = 90o. 9

450

400 Sun

350

300

250

200 Redshift (Z)

150

100

50

0 0 45 90 135 φ (in degree) 180 225 270 315 360

FIG. 3. Shows the variation of redshift (Z) versus φ for Sun from φ = 0 to 360o, at fixed θ = 90o. 10

TABLE I. A list showing Mass (M), (rg), Radius (R), Rotation velocity (Ω), and Rotation parameter (a) for different millisecond pulsars and Sun. Data taken from various authors as cited in column 2.

S. No. Star M(MSun) rg(km) R(km) Ω(rad/s) a(km) 1 PSR J 1748-2446ad [17] 1.350 4.050 20.1 4.4985×103 2.42325 2 PSR B 1937+21 [18] 1.350 4.050 20.2 4.0334×103 2.19438 3 PSR J 1909-3744 [19] 1.438 4.314 31.1 2.1300×103 2.74687 4 PSR 1855+09 1.350 4.050 46.9 1.1849×103 3.47509 5 PSR J 0737-3039 A [20] 1.340 4.020 133.6 0.2766×103 6.58269 6 PSR 0531+21 1.350 4.050 164.4 0.1885×103 6.79287 7 PSR B 1534+12 [21] 1.340 4.020 165.9 0.1657×103 6.08070 8 PSR B 1913+16 1.440 4.320 10.0 0.1064×103 0.01418 9 Sun 1.000 3.000 0.7×106 2.5982×10−6 1.69749

π 3π π TABLE II. Calculated values of Redshift (Z) for φ = 0, 2 and 2 , at fixed θ = 2 , corresponding redshifted of Lyman - α line (emitted wavelength of 1215.668 A˚) and the coefficient of latitude dependence of redshift (κ) for different millisecond pulsars, in case of (+,+) or (-,-), for (a, θ) combination. R a π 3π π π o S. No. Pulsar β = γ = Z(φ = 0) Z(φ = ) Z(φ = ) λred(φ = ) κ(φ = ,θ = 0 ) rg rg 2 2 2 2 1 PSR J 1748-2446ad 4.96296 0.59833 0.118672212 1.668503982 1.233631446 3244.0148 0.418608 2 PSR B 1937+21 4.98765 0.54182 0.118053061 1.663551032 1.263311491 3237.9936 0.419269 3 PSR J 1909-3744 7.20909 0.63673 0.077429231 2.047082682 1.640305917 3704.2408 0.353403 4 PSR 1855+09 11.58025 0.85805 0.046167768 2.697244032 2.256888342 4494.6211 0.282891 5 PSR J 0737-3039 A 33.23383 1.63749 0.015392030 4.897889501 4.399200503 7169.8753 0.172153 6 PSR 0531+21 40.59259 1.67725 0.012549338 5.488016592 5.017013830 7887.2739 0.156059 7 PSR B 1534+12 41.26866 1.51261 0.012340052 5.556341395 5.126599154 7970.3342 0.154401 8 PSR B 1913+16 2.31481 0.00328 0.326861792 1.020381544 1.017122955 2456.1131 0.656737

π (φ, θ = ) ℜ 2

2 (1 rg )(Rsinφ)+ rg a (1 rg )(Rsinφ)+ rg a rg 2 2 rga r r 2 rga r r = (1 ) (r + a + )( − 2 ) + 2( )( − 2 ) − r − r rga 2 2 rg a r rg a 2 2 rg a s r (Rsinφ) + (r + a + r ) r (Rsinφ) + (r + a + r ) − − (72)

Now as the photon is emitted from the surface of the star (Pulsar or Sun), we can substitute the actual value for ‘r’, which is ‘R’. π (φ, θ = ) ℜ 2

2 (1 rg )(Rsinφ)+ rg a (1 rg )(Rsinφ)+ rg a rg 2 2 rg a R R 2 rga R R = (1 ) (R + a + )( − 2 ) + 2( )( − 2 ) − R − R rga 2 2 rg a R rga 2 2 rg a s R (Rsinφ) + (R + a + R ) R (Rsinφ) + (R + a + R ) − − (73) on simplifying

4 4 2 2 2 2 2 2 2 R(R (R rg)+ a (rg + R)+ a R(2R r ) 2aRrg(a + R(R rg))sinφ + R (rg R)(a + R(R rg))sin φ) − − g − − − − = 3 2 2 s (R + a (rg + R) aRrgsinφ) − (74) R a We also express all the length scales in unit of rg, by substituting; = β, and = γ in above equation (74). Thus rg rg after certain simplifications

π β( β + β2 + γ2)((1 + β)(β2 +2γ2) + (β 1)β2cos2φ 4βγsinφ) (φ, θ = )= − − − (75) ℜ 2 2(β3 + γ2 + βγ2 βγsinφ)2 s −

In above equation (75) β and γ are dimensionless quantities and the redshift factor (φ, θ = π ) will show ℜ 2 11 a periodic with respect to φ. As can be seen from equation (75) each pulsar can be characterized by only two parameters β and γ. In For carrying out some numerical calculations we our case, we make TABLE - II, listing β and γ values ˚ π consider a Lyman - α line (1215.668 A) emitted from of these eight pulsars along with Z(φ = 0,θ = 2 ), the surface of the star (Pulsar or Sun). π π 3π π Z(φ = 2 ,θ = 2 ), Z(φ = 2 ,θ = 2 ) and Lyman - α line π π For an asymptotic observer, the redshifted (λred) Lyman redshifted wavelength λred(φ = 2 ,θ = 2 ) values. - α line can be calculated as, In FIG.-1 and FIG.-2, we make a plot showing λem π λred = λem(Z +1)= (76) variation of Z(φ, θ = 2 ) with φ for eight pulsars (listed ℜ in TABLE - I). We are making plot here for (+,+) or Further for doing numerical calculations we have consid- (-,-), for (a, θ) combination. It may be noted that φ ered the Newtonian approximation of angular momentum is actually equal to Ωt, where t is the time measured in observers coordinate. The nature is clearly periodic (J), π 3π with different amplitudes at φ = 2 and 2 . Thus for 2MR2Ω a pulsar as it rotates the redshifted values of spectrum J = (77) 5 shows a periodicity. From FIG.-1, FIG.-2 and TABLE - II, it is clearly seen that the value of redshift (Z) for So rotation parameter can be written as these millisecond pulsars obtains a maximum (primary maxima) at φ = π ,θ = π and a second maximum 2 2 2 2R Ω (secondary maxima with value less than the primary a = (78) 3π π 5c maxima) at φ = 2 ,θ = 2 . This asymmetry happens clearly due to the presence of sinφ and sin2φ terms where Ω is angular velocity of rotating body (Pulsar or in the expression of as in equations (72), (73) and Sun) as described in Section II A. There are two possi- (74). The value of redshift,ℜ at φ = 0,θ = π is actually bilities for direction of spin ( Ω) of central body. 2 same as that value of gravitational redshift which is We have four cases for the rotation± parameter (a) and the generally referred (in most of the published work) as position (θ) combination on the central body at which the gravitational redshift in Kerr field. The redshifted photon is emitted; π π wavelength value λred(φ = 2 ,θ = 2 ) is also given in a = +ive , θ = +ive , (+, +) TABLE - II. This calculation have been done for Lyman • ≡ - α line. For comparison we make a similar plot (FIG.-3) a = -ive , θ = -ive , ( , ) for Sun and the effect is too small. • ≡ − − a = +ive , θ = -ive , (+, ) • ≡ − a = -ive , θ = +ive , ( , +) • ≡ − IV. THE COEFFICIENT OF LATITUDE The Mass (M), Physical radius (R), Schwarzschild radius DEPENDENCE OF REDSHIFT rg, Rotation velocity (Ω), and Rotation parameter (a) π of Pulsars and Sun are calculated from different sources Below we define a parameter κ(φ = 2 ,θ), showing lat- in literature and these are listed in TABLE - I. itude dependence of redshift. From equation (76), we can write

λred,φ= π ,θ (Z + 1)φ= π ,θ φ= π ,θ= π π 2 = 2 = ℜ 2 2 = κ(φ = ,θ)(say) (79) λred,φ π ,θ π (Z + 1)φ π ,θ π φ π ,θ 2 = 2 = 2 = 2 = 2 ℜ = 2 π The quantity κ(φ = 2 ,θ) thus, is defined as red-shifted wavelength ratio at any value of latitude on the central body (star) at which photon is emitted and at equator. This parameter can be termed as ‘coefficient of latitude dependence of redshift from a rotating star’. Using the value of from equation (69) in above equation (79), we can write ℜ dφ dφ 2 π π [ gtt + gφφ( cdt ) +2gtφ( cdt )]θ= 2 κ(φ = ,θ)= (80) 2 q dφ 2 dφ [ gtt + gφφ( cdt ) +2gtφ( cdt )]θ q dφ π Here we can substitute the value of cdt (φ = 2 ,θ) from equation (48) and gtt, gφφ and gtφ from equations (3), (6) and dφ π π (7) for any value of θ. We can also substitute the value of cdt (φ = 2 ,θ = 2 ) from equation (49) and gtt, gφφ and gtφ π from equations (10), (11) and (12) for θ = 2 . 12

1.2 1.2 1.15 PSR J 1748-2446ad 1.15 PSR B 1937+21 1.1 1.1 1.05 1.05 1

1 κ(θ) κ(θ) 0.95 0.95 0.9 0.9 0.85 0.85 0.8

0.8 0 45 90 -90 -45 0 45 90 -90 -45 θ (in degree) θ (in degree)

1.2 1.15 1.15 PSR J 1909-3744 1.1 PSR 1855+09 1.1 1.05 1.05 1 κ(θ) 1 κ(θ) 0.95 0.95 0.9 0.9 0.85 0.85 0 0 45 90 45 90 -90 -45 -90 -45 θ (in degree) θ (in degree)

FIG. 4. Shows the variation of the coefficient of latitude dependence of redshift κ(θ) versus (θ), when rotation parameter (a) and θ having same sign (+,+) or (-,-) and opposite sign (+,-) or (-,+) for pulsars 1-4 (listed in TABLE - I) from θ = −90o to 90o, at fixed φ = 90o. The solid line represents the case for (+,+) or (-,-) and the dotted line represents the case for (+,-) or (-,+), for (a, θ) combination. 13

1.1 1.08 1.08 PSR J 0737-3039 A 1.06 PSR 0531+21 1.06 1.04 1.04 1.02 1.02 1

1 κ(θ) κ(θ) 0.98 0.98 0.96 0.96 0.94 0.94 0.92 0.92

0.9 0 45 90 -90 -45 0 45 90 -90 -45 θ (in degree) θ (in degree)

1.08 1.002 1.06 PSR B 1534+12 1.0015 PSR B 1913+16 1.04 1.001 1.02 1.0005 1 1 κ(θ) 0.98 κ(θ) 0.9995 0.96 0.999 0.94 0.9985 0.92 0.998 0 0 45 90 45 90 -90 -45 -90 -45 θ (in degree) θ (in degree)

FIG. 5. Shows the variation of the coefficient of latitude dependence of redshift κ(θ) versus (θ), when rotation parameter (a) and θ having same sign (+,+) or (-,-) and opposite sign (+,-) or (-,+) for pulsars 5-8 (listed in TABLE - I) from θ = −90o to 90o, at fixed φ = 90o. The solid line represents the case for (+,+) or (-,-) and the dotted line represents the case for (+,-) or (-,+), for (a, θ) combination. 14

1.000000 0.999999 0.999999 Sun 0.999998 0.999998 0.999997 κ(θ) 0.999997 0.999996 0.999996 0.999995 0.999995 0 45 90 -90 -45 θ (in degree) 1.000005 1.000005 1.000004 Sun 1.000004 1.000003 1.000003 κ(θ) 1.000002 1.000002 1.000001 1.000001 1.000000 0 45 90 -90 -45 θ (in degree)

FIG. 6. Shows the variation of the coefficient of latitude dependence of redshift κ(θ) versus (θ), when rotation parameter (a) and θ having same sign (+,+) or (-,-) and opposite sign (+,-) or (-,+) for Sun from θ = −90o to 90o, at fixed φ = 90o. The solid line represents the case for (+,+) or (-,-) and the dotted line represents the case for (+,-) or (-,+), for (a, θ) combination. 15

As a result we can write π κ(φ = ,θ)= 2

2 rg rg a rg rg a rg 2 2 rga R(1− r )+ r 2 rg a R(1− r )+ r 2 2 (1 r ) (r + a + r )( rg a rg a ) + 2( r )( rg a rg a ) − − − R+(r2+a2+ ) − R+(r2+a2+ ) r r r r r rg r rgra rgr rg ra 2 2 R (1− )+ 2 R (1− )+ rg r 2 2 rgra sin θ 2 sinθ ρ2 ρ2 2 asin θrgr sinθ ρ2 ρ2 (1 2 ) [r + a + ]sin θ( 2 ) + 2( 2 )( 2 ) ρ 2 rg ra 2 2 rgra 2 ρ rgra 2 2 rg ra 2 − − ρ − 2 (Rsinθ)+(r +a + 2 sin θ) − 2 (Rsinθ)+(r +a + 2 sin θ) s ρ ρ ρ ρ (81)

The denominator of the fraction in above equation (81) in case of (+,+) or (-,-) and similarly the same results π is redshift factor (φ = 2 ,θ), which clearly depends in case of (+,-) or (-,+), for (a, θ) combination. on the direction ofℜ spin ( a) of central body and also Finally we have only two situations for the combination on the position ( θ) on the± central body at which the of rotation parameter (a) and the position (θ) on the photon is emitted.± central body at which photon is emitted; (+,+) or (-,-) π This also indicates that κ(φ = 2 ,θ) will also depend on and (+,-) or (-,+), for (a, θ) combination. the position ( θ) and rotation parameter ( a). ± ± π We can find κ(φ = 2 ,θ) for θ = 0, which will give π We obtain same results for redshift (Z) and κ(φ = 2 ,θ) a ratio of redshifted wavelength between pole and equatorial region.

dφ dφ 2 π π [ gtt + gφφ( cdt ) +2gtφ( cdt )]θ= 2 κ(φ = ,θ =0)= (82) 2 q dφ 2 dφ [ gtt + gφφ( cdt ) +2gtφ( cdt )]θ=0 q rg r At poles (θ = 0), we use the values gtt = (1 2 2 ) and gφφ = gtφ = 0, from equations (13) to (15). Therefore, − r +a equation (82) can be rewritten as

dφ dφ 2 π π [ gtt + gφφ( cdt ) +2gtφ( cdt )]θ= 2 κ(φ = ,θ =0)= (83) 2 q rgr (1 2 2 ) − r +a q dφ π π Substituting the value of cdt (φ = 2 ,θ = 2 ) from equation (49) and gtt, gφφ and gtφ from equations (10), (11) and π (12) for θ = 2 , above equation (83) can be written as

2 rg rg a rg rg a rg 2 2 rga R(1− r )+ r 2 rg a R(1− r )+ r 2 2 (1 r ) (r + a + r )( rg a rg a ) + 2( r )( rg a rg a ) π − − − R+(r2+a2+ ) − R+(r2+a2+ ) κ(φ = ,θ =0)= r r r r r (84) 2 rg r (1 2 2 ) − r +a q

π π o In FIG.-4 and 5, we show κ(φ = 2 ,θ) for eight pulsars ues of κ(φ = 2 ,θ = 0 ) for these pulsars are also given listed in TABLE - I and II as a function of θ. It is seen in TABLE - II, for (+,+) or (-,-) combination. We make π that the value of κ(φ = 2 ,θ) is less than one for (+,+) FIG.-6, for Sun showing κ(θ) versus θ, for the purpose of or (-,-) combination while greater than one for (+,-) or comparison and we notice that the effect is too small. (-,+), for (a, θ) combination. This clearly indicates that the gravitational redshift increases from pole to equato- rial region (maximum at equator). In case of (+,+) or V. CONCLUSIONS (-,-), for (a, θ) combination. Again it decreases from pole to equatorial region (minimum at equator), in case of (+,- We can conclude from the present work that, ) or (-,+), for (a, θ) combination. This shows clearly the dependence of redshift on the latitude (90o θ). The val- − 1. Gravitational redshift is affected by rotation of the 16

central body. 7. Gravitational redshift decreases from pole to equa- torial region (minimum at equator), when rotation 2. Gravitational redshift is a function of the φ and parameter (a) and the position (θ) on the central latitude (90o θ). body at which the photon is emitted having oppo- − site signs (+,-) or (-,+), for (a, θ) combination. 3. Gravitational redshift will depend on the direction of spin ( a) of central body and also on the position 8. The coefficient of latitude dependence of redshift ( θ) on± the central body at which the photon is (κ), when the photon is emitted from any value of emitted.± latitude on the central body and equator, clearly shows the dependence of redshift on the latitude 4. When we consider the rotation velocity of central (90o θ). body (Ω) to be zero, then we can obtain the corre- − sponding gravitational redshift from a static body 9. Looking at the observed redshift from a rotating of same mass (Schwarzschild Mass). object it should be possible to comment, from what latitude (90o θ) of the object, the Lyman - α line − 5. Gravitational redshift shows a periodic nature with has been radiated. respect to φ. The value of redshift (Z) obtains a π π maximum (primary maxima) at, φ = 2 ,θ = 2 and a second maximum (secondary maxima with value ACKNOWLEDGMENTS 3π π less than the primary maxima) at φ = 2 ,θ = 2 . We wish to thank Dr. Atri Deshmukhya, Head De- 6. Gravitational redshift increases from pole to equa- partment of Physics, Assam University, Silchar, India torial region (maximum at equator), when rotation for inspiring discussions. A K Dubey is also thankful parameter (a) and the position (θ) on the central to Sanjib Deb and Arindwam Chakraborty, Department body at which the photon is emitted having same of Physics, Assam University, for providing help and sup- signs (+,+) or (-,-), for (a, θ) combination. port in programming and plots.

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