Scaling and Orbits for an Isotropic Metric

Conventional interpretation of the Einstein Equation has inconsistencies and contradictions, such as gravitational fields without energy, objects crossing event-horizons, objects exceeding the speed of light, and inconsistency in scaling the speed of light and its factors. An isotropic metric resolves such problems by attributing energy to the gravitational field, in the Einstein Equation. This paper discusses symmetries of an isotropic metric, including scaling of physical quantities, the Lorentz transformation, covariant derivatives, and stress-energy tensors, and transitivity of this scaling between inertial reference frames. Force, charge, Planck’s constant, and the fine structure constant remain invariant under isotropic gravitational scaling. Gravitational scattering, orbital period, and precession distinguish between isotropic and Schwarzschild metrics. An isotropic metric accommodates quantum mechanics and improves models of black-holes.


Introduction
The present interpretation of the Einstein Equation 8 G GT

G
in general relativity has troubling inconsistencies and contradictions, such as violation of semiclassical locality, quantum unitarity, time reversibility, and energy conservation [1].For example, when an object crosses a Swarzschild black-hole's event-horizon, it attains the speed of light, giving the object unbounded energy for nearby observers.Apparently, conservation of energy must be grossly violated, at least for local observers near the event-horizon.Since the Einstein Equation explicitly conserves energy, then the Einstein Equation must not work for local observers.Conventionally, one assumes that the Einstein Equation works only for distant observers, but by their location within the massive cosmos, all physical observers are local observers.So, the present interpretation of the Einstein Equation does not work for any physical observer.Moreover, a rotating black-hole can have a naked singularity, resulting in contradictions of time-travel [2].Since the conventional model of a black-hole predicts objects to enter a region where the model no longer makes sense, then something must be lacking from the model.
Since the Einstein Equation is designed to conserve energy, the failure to conserve energy must involve application of the equation, such as the failure to account for the energy density of the gravitational field.When one assembles electric charges on a sphere, one applies a force through a distance on the charges, and thus puts energy into the electric field.For gravity, ordinary mass plays the role of charge.When one assembles a sphere of mass, energy is released.So, a gravitational field should have negative energy density.The Einstein Equation equates   T , which is a contraction of the curvature tensor for space-time, to  0   d , which is the local energy and momentum density.The fact that the conventional Schwarzschild metric for a black-hole is derived by solving the differential equations for G for all regions around the singularity, implies that the gravitational fields have no energy nor momentum.
The resulting Schwarzschild metric for a black-hole is anisotropic: While objects in the gravitational well look shorter in a radial direction, their azimuthal dimensions remain unaffected, as viewed by a remote observer.Then, the speed of light is also anisotropic, and one cannot consistently scale mass and energy, and complications arise in reconciling gravity with quantum mechanics.These contradictions and inconsistencies should inspire us to consider a different metric.

Isotropic Metric
An isotropic metric with the scaling for time reciprocal that for space, yields a distance differential  , in terms of a distant observer's coordinates : The speed of light as seen from a distance,  , in terms of that locally, , is c x t x g g , which means that since , light in a gravitational well moves more slowly.As a result, a gravitational field deflects light.Therefore, this metric is not "conformally flat".Because the metric is isotropic, objects no longer cross eventhorizons.For example, one can see that in the orbit Equation (1.44) below, for a spherically symmetric potential, is bounded.In matrix form for spherical coordinates, this isotropic metric is so that the length differential is that in Equation (1.1).The term of the Einstein Tensor equals the total energy density.For an isotropic metric, it has two terms, one that has the form of a charge density, and the other that has the form of an energy density of a field [3]: where G is the gravitational potential in terms of scale factor g .One should ascribe the energy density of ordinary matter to the first term of the Einstein Tensor, and the energy density of the gravitational field to the second term.In the course of deriving this form, one finds that the metric scales momentum-energy like it scales space-time.Mass differential above.
Explicit inclusion of factors of c helps to verify scaling factors of g in these equations.Unlike the isotropic metric in Equation (1.1), isotropic metrics rejected in the past were conformally flat.They also did not include the corresponding relation (1.5) for momentum and energy, nor the energy of gravitational fields in the Einstein Tensor (1.4).While the same isotropic metric by Yilmaz [4] has an implicit globally preferred reference frame due to flawed assumptions ancillary to the form of the metric, the gravitational fields for the metric here have rest frames that vary from point to point, as shown in Equation (1.21) below.Such rest frames are consistent with frame-dragging.The "Parameterized Post-Newtonian" (PPN) parameters for equation (0.1) as defined on pp.1084-1085 of Gravitation [5] are: to first order in the gravitational constant , but differs greatly in the strong field limit.For example, the event-horizon is at  for an isotropic metric.So, one must look at strong gravitational fields to distinguish between them.

Scaling of Physical Quantities
It would help to consider scaling of physical quantities, to avoid blunders in gravitational scaling, and to identify those quantities that are invariant.Suppose a local observer in a gravitational field measures the distance between events, and a remote observer external to the gravitational field measures the distance between the same two events.In local coordinates, r and In terms of remote coordinates, . These substitutions into Equation (1.7) give the distance differential (1.1) from which one may infer the metric tensor, and calculate the affine connection and Einstein Tensor.Substitution , which shows that scaling is transitive for successive reference frames: Scale factor g grows to values greater than one, toward an attractive gravitational potential shows that, as seen by a remote observer, an object in a gravitational well is shorter.For time,

; energy density
; and mass, .Force and angular momentum are invariant.So, all observers agree on the value of .
The gravitational constant scales as .To change the scaling of a physical quantity, one can multiply by powers of . For example, the dimensionless quantity in g ,

Scaling in a Lorent Transformation
metric is Lorentz invariant reference frame, where scaling 1 g  , it is not L invariant for all observers.With proper choice of dinates, a Lorentz transformation of a vector where the Lorentz scaling is To simplify display in the rest of this section, dimensions not affected by a Lorentz transformati displayed.Then a Lorentz transformation and its inverse ar While the metric is not Lorentz invariant for all observers, the length differential is, w form Insertion of the Lorentz transformation and its inverse shows the Lorentz invariance of the length differe 1 0 where in the last step, the scaling transfers from the coordinates to the metric tensor.Insertion of a Lo formation and its inverse, in remote coordinates, shows that the Lorentz transformed metric is yielding a length differential in expected form in remote coordinates, Under a Lorentz transformation, the metric loses its isotropy, and acquires off-diagonal terms.Diagonalization generates the Lorentz transformation back to the local rest frame of the gravitational field, where the metric scaling factor g appears like an eigenvalue.

and ation of A an additional apparent transform
 in the , where the h C ristoffel symbol for the affine connection is Copyright © 2013 SciRes.

JMP
For example, the riant derivative in the y-direction of a vector along the cova If one divides through by 2   (1.25) th where the terms are scaled to the local observer's coordinates, plus a rotation: .
To preserve the symmetry of Equation (

, as sho in Eq
Therefore, to get the expected scaling for M  , one should de by a factor of also divi 2 c c g   .

Orbit Equation for an Isotropic Metric
Recently, long-lived stars have been found orbiting the black-hole in the center of the Milky y galaxy, in unexpectedly small orbit Wa s [6,7].Measurement of the ish precession of orbits might make it possible to distingu between a Schwarzschild metric and an isotropic metric, especially if one can observe an orbit smaller than the Schwarzschild radius.Furthermore, for an isotropic metric, non-decaying orbits exist at all distances from a black-hole.
As usual, equations of motion are Integration yields a constant of motion

  
2 , g m r Again, scaling for this constant is reconciled to invariant angular momentum, . While all observers agree on th momentum varies over its orbit, since its rest mass does not change verses different values of the metric.at e angular momentum of a particle, the contribution of the particle to the total angular while it tra- , .d Integration of this equation gives which is an expansion of Eq 7), to first order in G .The last term comes from the har- monic expansion Expansion (1.45) first differs from that for a Schwarzschild metric in the factor of two on the last term.So, orb warzschild metric.Since, 0 L m r (1.46) correction to the ital period for an isotropic metric is twice that for a Sch , where A , B , and Comparison of the above two equations shows that is 50% la schild metric.The com tation of rger than that given for a Schwarz parison would be complicated by ro the gravitational field.
For scattering, 1 0 r  .For a massless particle at this limit, , where r  is small deflection, i the impact parameter and v  is the initial speed.For a n this limit, 2 sin for a Schwarzschild metric.

Inconsistency for a Schwarzschild Metric
The Schwarzschild metric has inherent inconsistencies, ale the speed of light.For where c is one.The standard interpretation assumes that all observers agree on the metric.Therefore To relate the gravitational potential energy of a system, as measured locally, to that measured remotely, suppose, as an ansatz, that energy scales as (1.57)where is yet to be determined.Then has the same problem.Therefore the Sc ot use potenconserve momentum and energy in any physical reference frame.In contrast, an isotropic metric has selfconsistency across all inertial frames of reference, as shown by Equation (1.9).

Conclusion
That an isotropic metric accounts for the energy of a gravitational field, should be sufficient reason to adopt an isotropic metric over a S isotropic metric, su tial and the mass-energy-momentum equation.The invariance under isotropic scaling of force, angular momentum, electric field, electric charge, and fine structure constant provide consistency of general relativity with both quantum mechanics and electromagnetism.Orbits no longer cross event horizons.Inconsistencies in scaling for a Schwarzschild metric make the Schwarzschild metric untenable, necessitating adoption of the isotropic metric.


are all the same Lorentz transformation.Then the Lorentz transformed metric 2 1 a curved orbit, while that in (1.1) describes a straight-line distance.From (1.1), ident in (1.1), but the scaling in (1same agnitude.The orbiter's ene y and the remote observer's metric determine co of motion k unprimed are local measures.For a point source, the Schwarzschild metric scaling

.
tric implies a preferred remote frame of reference in which physics is self-consistent; one cann tials to chwarzschild metric.Further reason is provided by the symmetries of scaling for an ch as that between the length differen-NCES day, Vol.66, 2013, pp.30-35.
In the same way, any other derivative with respect to time sh uld be divided by a factor of c , to scale it like derivatives with respect to ace.Then, 38)