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Gravitational spectral shift exterior to the sun, earth and the other oblate spheroidal planets.

1 Introduction

According to the General Theory of Gravitation, the rate of a clock is slowed down when it is in the vicinity of a large gravitating mass. Since the characteristic frequencies of atomic transitions are, in effect, clocks, one has the result that the frequency of such a transition occurring, say, on the surface of the Sun, should be lowered by comparison with a similar transition observed in a terrestrial laboratory. This manifests itself as a gravitational red shift in the wavelengths of spectral lines [1]. It has been experimentally and astrophysically observed that there is an increase in the frequency of light (photon) when the source or emitter is further away from the body than the receiver. The frequency of light will increase (shifting visible light towards the blue end of the spectrum) as it moves to lower gravitational potentials (into a gravity well). Also, there is a reduction in the frequency of light when the source or emitter is nearer the body than the receiver. The frequency of light will decrease (shifting visible light towards the red end of the spectrum) as it moves into higher gravitational potentials (out of a gravity well). This was experimentally confirmed in the laboratory by the Pound-Rebka experiment in 1959 (they used the Mossbauer effect to measure the change in frequency in gamma rays as they travelled from the ground to the top of Jefferson Labs at Havard University) [2]. This gravitational phenomenon was later confirmed by astronomical observations [3]. In this article, we verify the validity of our metric tensor exterior to a massive homogeneous oblate spheroid by studying gravitational spectral shift in the vicinity of the Sun, Earth and other oblate spheroidal planets. Basically, we assume that these gravitational sources are time independent and homogeneous distributions of mass within spheroids, characterized by at most two typical integrals of geodesic motion, namely, energy and angular momentum. From an astrophysical point of view, such an assumption, although not necessary, could, however, prove useful, because it is equivalent to the assumption that the gravitational source is changing slowly in time so that partial time derivatives are negligible compared to the spatial ones. We stress that the mass source considered is not the most arbitrary one from a theoretical point of view, but on the other hand, many astrophysically interesting systems are usually assumed to be time independent (or static from another point of view) and axially symmetric continuous sources.

2 Covariant metric tensor exterior to a massive homogeneous oblate spheroid

The covariant metric tensor in the gravitational field of a homogeneous oblate spheroid in oblate spheroidal coordinates (n, <p) has been obtained [4, 5] as;

[g.sub.00] = (1 + 2/[c.sup.2] f([eta],[xi]), (2.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2-2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

[g.sub.33] = -[a.sup.2](1 + [[xi].sup.2])(1 - [[eta].sup.2]), (2.5)

f ([eta],[xi]) is an arbitrary function determined by the mass or pressure distribution and hence possesses all the symmetries of the latter, a priori. Let us now recall that for any gravitational field [4-7]

[g.sub.00] = 1 + 2/[c.sup.2] [PHI] (2.6)

where [PHI] is Newton's gravitational scalar potential for the field under consideration. Thus we can then deduce that the unknown function in our field equation can be given approximately as

f ([eta],[xi]) [congruent to] ([eta],[xi]), (2.7)

where [PHI] ([eta],[xi]) is Newton's gravitational scalar potential exterior to a homogeneous oblate spheroidal mass. It has been shown that [8];

[PHI] ([eta],[xi]) = [B.sub.0][Q.sub.0] (-i[xi]) [P.sub.0]([eta]) + [B.sub.2][Q.sub.2](-i[xi])[P.sub.2]([eta]), (2.8)

where [Q.sub.0] and [Q.sub.2] are the Legendre functions linearly independent to the Legendre polynomials [P.sub.0] and [P.sub.2] respectively; [B.sub.0] and [B.sub.2] are constants given by

[B.sub.0] = 4[pi]G[[rho].sub.0][a.sup.2][[xi].sub.0]/3[[DELTA].sub.1]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[DELTA].sub.1] and [[DELTA].sub.2] are defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

G is the universal gravitational constant, [[rho].sub.0] is the uniform density of the oblate spheroid and a is a constant parameter.

In a recent article [9], we obtained a satisfactory approximate expression for equation (2.8) as;

[PHI]([eta], [xi]) [approximate equal to] [B.sub.0]/3[[xi].sup.3] (1 + 3[[xi].sup.2]) i - [B.sub.2]/30[[xi].sup.3] (7 + 15[[xi].sup.2]) (3 [xi.sup.2]-i)i (2.9)

with

[PHI]([eta], [xi]) [approximate equal to] [B.sub.0]/3[[xi].sup.3] (1 + 3[[xi].sup.2]) i + [B.sub.2]/30[[xi].sup.3] (7 + 15[[xi].sup.2])i

and

[PHI]([eta], [xi]) [approximate equal to] [B.sub.0]/3[[xi].sup.3] (1 + 3[[xi].sup.2]) i - [B.sub.2]/15[[xi].sup.3] (7 + 15[[xi].sup.2])i

as the respective approximate expressions for the gravitational scalar potential along the equator and pole exterior to homogeneous oblate spheroidal bodies. These equations were used to compute approximate values for the gravitational scalar potential exterior to the Sun, Earth and other oblate spheroidal planets [9].

[FIGURE 1 OMITTED]

3 Gravitational spectral shift exterior to oblate spheroidal distributions of mass

Here, we consider a beam of light moving from a source or emitter at a fixed point in the gravitational field of the oblate spheroidal body to an observer or receiver at a fixed point in the same gravitational field. Einstein's equation of motion for a photon is used to derive an expression for the shift in frequency of a photon moving in the gravitational field of an oblate spheroidal mass.

Now, consider a beam of light moving from a source or emitter (E) at a fixed point in the gravitational field of an oblate spheroidal body to an observer or receiver (R) at a fixed point in the field as shown in Fig. 1.

Let the space time coordinates of the emitter and receiver be ([t.sub.E], [[eta].sub.E], [[xi].sub.E], [[phi].sub.E]) and ([t.sub.R], [[eta].sub.R], [[xi].sub.R], [[phi].sub.R]) respectively. It is well known that light moves along a null geodesic given by

d[tau] = o. (3.1)

Thus, the world line element for a photon (light) takes the form

[c.sup.2][g.sub.00] [dt.sup.2] = [g.sub.11] d[[eta].sup.2] + 2[g.sub.12][d.sub.[eta]]d[xi] + [g.sub.22]d[[xi].sub.2] + [g.sub.33] d[[phi].sup.2]. (3.2)

Substituting the covariant metric tensor for this gravitational field in equation (3.2) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

Now, let u be a suitable parameter that can be used to study the motion of a photon in this gravitational field. Then equation (3.3) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

Equation (3.4) can be equally written as

dt/du = 1/c [(1 + 2/[c.sup.2] f([eta],[xi]).sup.-1/2] ds, (3.5)

where ds is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

Integrating equation (3.5) for a signal of light moving from emitter E to receiver R gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

The time interval between emission and reception of all light signals is well known to be the same for all light signals in relativistic mechanics (constancy of the speed of light) and thus the integral on the right hand side is the same for all light signals. Consider two light signals designated 1 and 2 then

[t.sup.1.sub.R] - [t.sup.1.sub.E] = [t.sup.2.sub.R] - [t.sup.2.sub.E] (3.8)

or

[t.sup.2.sub.R] - [t.sup.1.sub.R] = [t.sup.2.sub.E] - [t.sup.1.sub.E] (3.9)

Thus,

[DELTA][t.sub.R] = [DELTA][t.sub.E]. (3.10)

Hence, coordinate time difference of two signals at the point of emission equals that at the point of reception. From our expression for gravitational time dilation in this gravitational field [10], we can write

[DELTA][t.sub.R] = [(1 + 2/[c.sup.2] [f.sub.R]([eta],[xi]).sup.1/2] [DELTA][t.sub.R]. (3.11)

Equations (3.9), (3.10) and (3.11) can be combined to give

[DELTA][[tau].sub.R]/[DELTA][[tau].sub.E] = [[(1 + 2/[c.sup.2] [f.sub.R]([eta],[xi])/1 + 2/[c.sup.2] [f.sub.E]([eta],[xi])]).sup.1/2]. (3.12)

Now, consider the emission of a peak or crest of light wave as one event. Let n be the number of peaks emitted in a proper time interval [DELTA][[tau].sub.E], then, by definition, the frequency of the light relative to the emitter, [v.sub.E], is given as

[v.sub.E] = n/[DELTA][[tau].sub.E]. (3.13)

Similarly, since the number of cycles is invariant, the frequency of light relative to the receiver, [v.sub.R], is given as

[v.sub.R] = n/[DELTA][[tau].sub.R]. (3.14)

Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3-15)

or

[v.sub.R]/[v.sub.E] [approximately equal to] (1 + 2/[c.sup.2] [f.sub.E]([eta],[xi]))(1 - 2/[c.sup.2] [f.sub.R] ([eta],[xi]) (3-16)

or

vR/vE - 1 [approximately equal to] - 1/[c.sup.2] [f.sub.E]([eta],[xi]) - [f.sub.R]([eta],[xi] (3.17)

to the order of [c.sup.-2]. Alternatively, equation (3.17) can be written as

z [equivalent to] = [DELTA]v/[v.sub.E] [equivalent to] [[v.sub.R] - [v.sub.E]/[v.sub.E]] [approximately equal to] 1/[c.sup.2] [f.sub.E]([eta],[xi]) - [f.sub.R]([eta],[xi]]. (3.18)

It follows from equation (3.18) that if the source is nearer the body than the receiver then [f.sub.E]([eta], [xi]) < [f.sub.R]([eta], [xi]) and hence [DELTA]v < 0. This indicates that there is a reduction in the frequency of light when the source or emitter is nearer the body than the receiver. The light is said to have undergone a red shift (that is the light moves toward red in the visible spectrum). Otherwise (source further away from body than receiver), the light undergoes a blue shift. Now, consider a signal of light emitted and received along the equator of the homogeneous oblate spheroidal Earth (approximate gravitational field where f([eta],[xi]) [approximately equal to] [PHI]([eta],[xi]). The ratio of the shift in frequency to the frequency of the emitted light at various points along the equator and received on the equator at the surface of the homogeneous oblate spheroidal Earth can be computed using equation (3.18). This yields Table 1. Also, the ratio of the shift in frequency of light to the frequency of the emitted light on the equator at the surface and received at various points along the equator of the homogeneous oblate spheroidal Earth can be computed. This gives Table 2.

Tables 1, thus confirms our assertion above that there is an increase in the frequency of light when the source or emitter is further away from the body than the receiver. The frequency of light will increase (shifting visible light toward the blue end of the spectrum) as it moves to lower gravitational potentials (into a gravity well). Table 2, also confirms our assertion above that there is a reduction in the frequency of light when the source or emitter is nearer the body than the receiver. The frequency of light will decrease (shifting visible light toward the red end of the spectrum) as it moves to higher gravitational potentials (out of a gravity well). Also, notice that the shift in both cases increases with increase in the distance of separation between the emitter and receiver. The value of the shift is equal in magnitude at the same separation distances for both cases depicted in Tables 1 and 2.

Now, suppose the Pound-Rebka experiment is performed at the surface of the Sun, Earth and other oblate spheroidal planets on the equator. Then, since the gamma ray frequency shift was observed at a height of 22.5m above the surface, we model our theoretical computation and calculate the theoretical value for this shift. This computation yields Table 3.

With these predictions, experimental astrophysicists and astronomers can now attempt carrying out similar experiments on these bodies. Although, the prospects of carrying out such experiments on the surface of some of the planets and Sun are less likely (due to temperatures on their surfaces and other factors); theoretical studies of this type helps us to understand the behavior of photons as they leave or approach these astrophysical bodies. This will thus aid in the development of future instruments that can be used to study these heavenly bodies.

4 Conclusion

The practicability of the findings in this work is an encouraging factor. More so, that in this age of computational precision, the applications of these results is another factor.

Submitted on June 01, 2010 / Accepted on June 05, 2010

References

[1.] Matolcsi T. and Matolcsi M. GPS revisited: the relation of proper time and coordinate time. arXiv: math-ph/0611086.

[2.] Pound R.V. and Rebka G.A. Jr. Gravitational red shift in nuclear resonance. Physical Review Letters, 1959, v. 3(9), 439-441.

[3.] Ohanian H.C. and Remo R. Gravitation and space-time. W.W. Norton and Company, 1994.

[4.] Howusu S.X.K. The 210 astrophysical solutions plus 210 cosmological solutions of Einstein's gravitational field equations. Natural Philosophy Society, Jos, 2007, 47-79.

[5.] Chifu E.N., Usman A. and Meludu O.C. Orbits in homogeneous oblate spheroidal gravitational space-time. Progress in Physics, 2009, v.3, 49-53.

[6.] Chifu E.N. and Howusu S.X.K. Gravitational radiation and propagation field equation exterior to astrophysically real or hypothetical time varying distributions of mass within regions of spherical geometry. Physics Essays, 2009, v.22(1), 73-77.

[7.] Chifu E.N. and Howusu S.X.K. Solution of Einstein's geometrical field equations exterior to astrophysically real or hypothetical time varying distributions of mass within regions of spherical geometry. Progress in Physics, 2009, v.3, 45-48.

[8.] Howusu S.X.K. Gravitational fields of spheroidal bodiesextension of gravitational fields of spherical bodies. Galilean Electrodynamics, 2005, v.16(5), 98-100.

[9.] Chifu E.N., Usman A., and Meludu O.C. Gravitational scalar potential values exterior to the Sun and planets. Pacific Journal of Science and Technology, 2009, v.10(1), 663-673.

[10.] Chifu E.N., Usman A., and Meludu O.C. Gravitational time dilation and length contraction in fields exterior to static oblate spheroidal mass distributions. Journal of the Nigerian Association of Mathematical Physics, 2009, v.15, 247-252.

Chifu Ebenezer Ndikilar *, Adam Usman ([dagger]), and Osita C. Meludu ([dagger])

* Department of Physics, Gombe State University, P.M.B 127, Gombe, Nigeria

([dagger]) Department of Physics, Federal University of Technology, Yola, Adamawa State, Nigeria

E-mails: ebenechifu@yahoo.com; aausman@yahoo.co.uk; ocmeludu@yahoo.co.uk
Fig. 2: Ratio of the shift in frequency of light to the
frequency of the emitted light at points along equator and
received on the surface of the Earth on the equator.

Emi Pt        Recep pt      z (x[10.sup.-10])    Type of
                                                  shift

[xi]0       [[xi].sub.0]            0              none
2 [xi]t     [[xi].sub.0]         3.454804          blue
3 [xi]t     [[xi].sub.0]         4.603165          blue
4 [xi]t     [[xi].sub.0]         5.176987          blue
5 [xi]t     [[xi].sub.0]         5.521197          blue
6 [xi]t     [[xi].sub.0]         5.750643          blue
7 [xi]t     [[xi].sub.0]         5.914522          blue
8 [xi]t     [[xi].sub.0]         6.037426          blue
9 [xi]t     [[xi].sub.0]         6.133016          blue
10 [xi]t    [[xi].sub.0]         6.209486          blue

Fig. 3: Ratio of the shift in frequency of light to the
frequency of the emitted light at points along equator and
received on the surface of the Earth on the equator.

Emi Pt             Recep pt       z (x[10.sup.-10])    Type of
                                                        shift

[[xi].sub.0]     [[xi].sub.0]             0              none
[[xi].sub.0]    2 [[xi].sub.0]        -3.454804          red
[[xi].sub.0]    3 [[xi].sub.0]        -4.603165          red
[[xi].sub.0]    4 [[xi].sub.0]        -5.176987          red
[[xi].sub.0]    5 [[xi].sub.0]        -5.521197          red
[[xi].sub.0]    6 [[xi].sub.0]        -5.750643          red
[[xi].sub.0]    7 [[xi].sub.0]        -5.914522          red
[[xi].sub.0]    8 [[xi].sub.0]        -6.037426          red
[[xi].sub.0]    9 [[xi].sub.0]        -6.133016          red
[[xi].sub.0]   10 [[xi].sub.0]        -6.209486          red

Fig. 4: Predicted Pound-Rebka shift in frequency along the equator
for the Sun, Earth and the other oblate spheroidal planets.

Body          Radial       [xi]           [[PHI].sub.E]
              dist.       at pt         (Nm [kg.sup.-1])
               (km)

Sun         700,022.5    241.527    -1.9375791 x [10.sup.11]
Earth       6,378.023     12.010     -6.2079113 x [10.sup.7]
Mars         3,418.5      9.231      -1.2401149 x [10.sup.7]
Jupiter      71,512.5     2.641      -1.4968068 x [10.sup.9]
Saturn       60,292.5     1.971      -4.8486581 x [10.sup.8]
Uranus       25,582.5     3.994      -2.1563913 x [10.sup.8]
Neptune      24,782.5     4.304      -2.5243240 x [10.sup.8]

Body              [[PHI].sub.R]                 Predicted
                (Nm [kg.sup.-1])                  shift

Sun         -1.9373218 x [10.sup.11]     -2.85889 x [10.sup.-21]
Earth        -6.2078881 x[10.sup.7]      -2.57800 x [10.sup.-15]
Mars         -1.2317966 x[10.sup.7]      -9.24256 x [10.sup.-20]
Jupiter      -1.4958977 x[10.sup.9]     -1.010111 x [10.sup.-20]
Saturn       -4.8484869 x[10.sup.8]     -1.902222 x [10.sup.-21]
Uranus       -2.1522082 x [10.sup.8]    -4.647889 x [10.sup.-20]
Neptune      -2.5196722 x [10.sup.8]    -5.168667 x [10.sup.-20]
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Author:Ndikilar, Chifu Ebenezer; Usman, Adam; Meludu, Osita C.
Publication:Progress in Physics
Date:Oct 1, 2010
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