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A Possible Scalar Term Describing Energy Density in the Gravitational FieldThe gravitational field of a point mass and the electric field of a point charge are structurally similar. Each may be represented by a vector field in which the vectors are directed along radial lines emanating from the point, and the vector magnitudes decrease as the inverse square of the [radial] distance from the point. The electric field, E, when multiplied by the magnitude of a test charge q in the field, gives the force f exerted locally on the test charge
The gravitational field, g, when multiplied by the magnitude of a test mass m in the field, gives the force f exerted locally on the test mass
In each case, the force has the same vector sense as the field. The electric field may point radially toward or away from the source charge, depending on the sign of the source charge. The gravitational field points toward the source mass in all known cases. The electric field has a scalar energy density field (or,following some older texts, a pressure field) associated with it. When the field vector at a point is E, then the energy density at the same point is where _{0}^{15} is the permittivity of free space. Energy density is a measure of the energy stored in the field per unit volume of space. Its unit of measure is j/m^{3} (or nt/m^{2}, if it is thought of as a pressure). While eq. 3 represents the energy density for the electric field, and a similar expression represents the energy density for the magnetic field, no such energy density term has ever been defined for the gravitational field. But one suspects that it could be, and possibly even should be. Let us use the similarity between the gravitational and electric fields to construct a gravitational energy density term. We begin by noting how the permittivity of free space enters into the expression for E:
is a universal constant, Q is the source charge, and r is radial distance from the source. The gravitational field is given by the expression
where G is a universal constant, M is the source mass, and r is radial distance from the source. The electrical field energy density may be written in terms of k as u_{E}
= ½ (1/4k)
E^{2}
By analogy, a candidate gravitational energy density term may now be constructed and written as Near the surface of the earth g =
9.807 m/sec^{2}.
Also G = 6.672 X 10^{-11} (nt m^{2})/kg^{2} so that u_{G} = 5.736 X 10^{10} j/m^{3}. Using eq. 7, it might be possible to construct a classical argument for the rotation of perihelion of a planet around its central body [sun]. Recall that the rotation of Mercury's perihelion (43 arc sec per century) was successfully dealt with by general relativity. Historically, attempts to modify Newton's law of gravitation to account for the observed motion of Mercury have proved unsatisfactory. So did the introduction of another [hypothetical] planet called Vulcan, with an orbit inside that of Mercury. It is reasonable, then, that the present argument will not be an exact solution, but a rough approximation. Its usefulness is as a conceptual tool for students, familiar with classical dynamics, who are just being introduced to Einstein's concepts. It will serve the function of a bridge between classical orbit theory and general relativity. Einstein's original paper on the deflection of starlight passing the sun might be said to serve a similar function. Einstein's calculation in that paper, is still reproduced as a problem in texts on special relativity. Throughout this calculation, we will use Newton's law of gravitation without modification, and incorporate the energy density term described above to render a qualitative description of the rotation of perihelion. First, we rewrite the term u_{G} by substituting for g in eq. 7.: g =
GM/r^{2}
Another scalar field may be obtained from the energy density field by using the mass-energy equivalence from special relativity; i.e., a scalar mass density field of the form We next assume that the mass due to the term u_{G}/c^{2}, integrated over a suitable volume of space, behaves, gravitationally, like ordinary matter. For an extended body like the sun (rather than an ideal point mass), the ramifications of this assumption need to be explored i. In the interior of the extended body, since the interior field contributes to the overall mass, and, therefore, to its gravitational field, and
The sun may be considered a sphere of radius r_{0}, and constant (average) density . The classical solar mass is then M_{0}
= 4r_{0}^{3}/3
To find the additional mass contribution due to the interior field, we must first rewrite eq. 9 for the sun's interior. We set M =
4r^{3}/3
which is a function of the radial distance r. We next substitute this result into eq. 9 to obtain The corresponding mass contribution Mf is then the volume integral of eq. 10 throughout the sun's interior. The integrand is dM_{f}
= [2Gr^{2}^{2}/(9c^{2})]
^{.} 4r^{2}
dr
and the limits of integration are from r = 0 to r = r_{0}. Thus
The total mass of the sun must now be
For the sun, we know that M_{0}
= 1.99 X 10^{30} kg
= 1,410 kg/m^{3} (avg.) r_{0} = 6.912 X 10^{8} m We calculate M_{f} = 4.08 X 10^{23} kg As should be expected, M_{f} is very small compared to M_{0}. In fact, M_{f}
= (2.05 X 10^{-7}) ^{.} M_{0}
and is, therefore, only slightly greater than one ten-millionth of the classical solar mass. Outside the sun, the local field experienced by an orbiting planet, asteroid, etc., has contributions from i. The mass M = M_{0} + M_{f} (a constant in this region), and
Let a planet be located a distance r from the center of the sun. Associate with this distance a sphere, concentric with the sun, on whose surface the planet is always to be found. As r increases, the sphere expands. As r decreases, the sphere contracts. The motion of the planet is determined by the total mass inside the sphere. This total mass is made up of the mass contained in the sun plus the mass due to the external field contained inside the sphere. For the external field dM_{f'}
= [GM^{2}/(8r^{4}c^{2})]
^{.} 4r^{2}
dr
The volume integral is to be taken throughout the region between the surface of the sun and the sphere of radius r. Thus It is the total mass M + M_{f'} (= M_{0} + M_{f} + M_{f'}) that attracts the planet, and influences its motion around the sun. The gravitational field g acting on the planet is This field represents the acceleration of the planet in its orbit around the sun. The classical contribution of the sun is represented by
and results in a stationary, elliptical orbit, as expected. The additional contribution of the sun, due to its external gravitational field, is represented by
This field varies with radial distance r, the more so the smaller the value of r. Therefore, any planet, near the sun and in a noncircular orbit, will experience a sideways perturbation from a strict classical orbit. If, without this perturbation, the planet would follow a classical ellipse, the perturbation must be such as to divert it sunwards from the ellipse as it travels from perihelion to aphelion. The reverse must be true for the other half of the orbit, with the result that the line of apsides must turn [slowly] with the same directional sense as the orbital motion of the planet (i.e., clockwise or counterclockwise, viewed from a suitable vantage point)^{16}. It is interesting to calculate the approximate magnitude of the mass contribution in eq. 13 for the planet Mercury. Accordingly, we need to know that for Mercury, r =
5.75 X 10^{10} m
Also, for the sun, M = 1.99 X 10^{30} kg r_{0} = 6.91 X 10^{8} m And G = 6.67 X 10^{-11} nt m^{2}/kg^{2}. Then, It is interesting that the mass calculated in eq. 17 is roughly equivalent to the mass of the earth. It was commented earlier that when Mercury's rotation of perihelion was first observed, astronomers attempted to account for it by postulating another planet called Vulcan, whose orbit was inside that of Mercury and whose gravitational influence perturbed Mercury's orbit. These observations were made late in the nineteenth century, before the advent of special relativity. Vulcan was never found, as we now know, and the perturbation in Mercury's orbit was finally accounted for satisfactorily by Einstein.
^{16} It may be shown that motion, under a law of central attraction of the general form results in the type of motion we have been describing here; i.e., that the perihelion of a planet, moving under such a law of attraction, will rotate at a rate proportional to the constant, k_{1} (Levi-Civita, pg. 396). The law of attraction stated in eq. 14a, above, only approximates eq. 18. We may rewrite eq. 14a. as G(M
+ GM^{2}(1/r_{0}-1/r)/(2c^{2}))/r^{2}
= GM/r^{2} + [G^{2}M^{2}(r/r_{0}-1)/2c^{2}r^{3})]
14b.
In this representation, comparing with eq. 18: k =
GM
k_{1} = G^{2}M^{2}(r/r_{0}-1)/(2c^{2}) The term k_{1} is not a constant, as eq. 18 requires, but is a linear function of the distance, r. Thus, our use of the energy density term provides, at best, only a crude approximation to the perihelion problem. None the less, it still seems reasonable that the approach described above should have instructional value, providing a conceptual bridge between the worlds of classical physics/special relativity on the one hand, and general relativity on the other.
Line of apsides: the line connecting the ascending and descending nodes of the orbit. |
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