RIEMANN'S GEOMETRY AND NEWTON'S GRAVITATION

James Constant

Constantrcs@cs.com

General Relativity (GR) is hard to reconcile with the rest of physics, and even within its own structure has weaknesses. Its heavy mathematical structure seems utterly incompatible with Quantum Mechanics and is at best tenuous with Special Relativity (SR) and Newton's Theory of Gravitation (NTG). Different as Einstein's and Newton's theories are, within the solar system, their results are almost identical. Yet, for over 80 years, the measurements are less than satisfactory and competing theories have emerged to explain the observations.¹ While the paucity and inconclusiveness of experimental evidence ² is an obstacle to the confirmation of general relativity, there are some other indicators. This page will examine some difficulties in reconciling GR with SR and NTG.

Special Relativity and Gravitation

Special relativity has to be reconciled with any theory of gravitation. We begin by considering the variation of mass with relative speed v

....................................................................................... (1)

in which m_o=rest mass and c=speed of light. In equation (1), when which holds in Newton's theory of gravitation. Also, when the Lorentz transformations of special relativity reduce to the Galilean transformations of Newton's theory of gravitation. Thus, special relativity easily reconciles with Newton's theory of gravitation. In general relativity, we assume that Riemann's geometry reduces to Newton's gravitation and thus, by proxy, that special relativity reconciles with general relativity.

The Principle of Equivalence and Special Relativity

Gravity was assumed by Newton to be propagated instantly at any distance. Einstein assumed a gravitation speed equal to that of light. It is remarkable that we know nothing experimentally about the speed of gravity. We can only fall back on a limited number of actual observations, the main one being Galileo's law which states that all bodies fall with equal acceleration. This law follows from Newton's second law of motion F=m_ia in which F is any type of force, gravitational or non-gravitational, m_i=inertial mass, and a=acceleration. If the force is a gravitational force F_g=m_ga_g and, therefore, m_ga_g=m_ia.

Non-relativistic experiments have established that the gravitational and inertial masses are equal, m_g=m_i to an accuracy of 1 part in 10¹³. Accordingly,

a_g=a ........ and ........ F_g=F ........................................ .................. (2)

Equations (2) state there is no way of telling between a non-relativistic gravitational and non-gravitational force, i.e., the gravitational field can be imitated or compensated by a non-gravitational force. This is what Einstein calls the "principle of equivalence" and forms the heart of his theory of gravitation. After all, if we cannot distinguish between gravitational and non-gravitational accelerations and forces, we can do away with gravitational forces and their bothersome action-at-a-distance and claim that non-gravitational forces, or even Riemann's curved geometry, describe reality.

While Einstein's principle of equivalence is confirmed by non-relativistic Newtonian experiments proving the equality of gravitational and inertial masses, equation (1) says it cannot be confirmed by relativistic experiments because these are expected to show the inequality of gravitational and non-gravitational accelerations and forces. For example, in view of (1), and since the relativistic form of Newton's second law of motion is , we find by expanding,

............................................................................. (3)

and, since m_g=m, F_g=m_ga_g, F=ma (stationary source a_g, relativistic mass m)

........ and ........ ........................................................ (4)

Equations (4) state there is a way of telling between a relativistic gravitational and non-gravitational force, i.e., the gravitational field cannot be imitated or compensated by a non-gravitational force. This means that Einstein's principle of equivalence fails at relativistic speeds. At these speeds, we can distinguish between gravitational and non-gravitational accelerations and forces, and we can no longer do away with gravitational forces and their bothersome action-at-a-distance and claim that non-gravitational forces, or even Riemann's curved geometry, describe reality. Of course, we must await the relativistic experiments to confirm these results. In the meantime, it is clear that Einstein's principle of equivalence, proven in the non-relativistic Newtonian realm, cannot be invoked to prove relativistic observations (classical tests, time delay, binary pulsars, etc.).

Einstein's Metrics and Newton's Potential

It is stated that a particle moving slowly in a weak stationary gravitational (Newtonian) field obtains an acceleration in which g_oo is Einstein's "gravitational potential" in the weak field low velocity case. The corresponding Newtonian result is in which is the Newtonian gravitational potential. We are then told ad hoc that³

provided g_oo .............................................................. (5)

We are further told ad hoc that⁴ and since

provided g_oo ........ and ........ ..................... (6)

Equations (5) and (6) say, first, that Einstein's component of the metric tensor g_oo is a "gravitational potential" which can be connected to Newton's potential and, second, that Einstein's energy density T_oo is Newton's mass density . In his paper, which is the fundamental statement of general relativity, Einstein (1916 Section 4) states that "the ten functions representing the gravitational field at the same time define the metrical properties of the space measured". This statement is presented as a property of nature which reduces gravitation to geometry, a genial mathematical work whose application to physics remains open to discussion.⁵ Thus, generally, Einstein's theory claims that the metric components g_ik describe a particular type of a non-gravitational field which, by invoking the principle of equivalence, constitutes a non-gravitational force replacing Newton's gravitational field and gravitational force. However, metric components g_ik are mathematical functions of the coordinates with no physical meaning. They do not inherently reduce to functions of physical gravitational potential . Nor does mathematical tensor component T_oo inherently reduce to physical mass density . These mathematical entities cannot be confirmed independently by experiment but receive status only because it is assumed they can be connected to physical Newtonian fields. Conditions g_oo and are, therefore, ad hoc. Einstein's metrics cannot be arbitrarily connected to Newton's potential and mass density . Without such ad hoc connections to Newtonian fields, the classical non-relativistic tests of Einstein's theory are inherently indeterminable by Einstein's metrics taken alone.

Einstein's Metrics and Boundary Conditions

In a study of a variety of physical problems one often arrives at Laplace's equation

.............................................................................................................................. (7)

The function u is said to be harmonic in the domain T if it is continuous in this region together with its derivatives up to a second order and if it satisfies Laplace's equation (7). In the presence of sources, we obtain Poisson's equation

..................................................................................................................... (8)

where f is the density of the source and r is radial distance.

If we consider some volume T, bounded by the surface , the problem of a steady distribution of u inside the volume T is formulated only if we specify the boundary condition. There are many boundary value problems some main ones being

1. Dirichlet's problem: u=f₁ at the surface

2. Newmann's problem: u=f₂ at the surface

3. External problem: at the surface

where f₁,f₂,f_3,h are given functions, is the derivative with respect to the outer normal to the surface . The physical significance of boundary conditions is obvious since a potential function is uniquely determined only when its boundary conditions are known.⁶ In Newton's theory, and (r)-->0 as . In Einstein's theory, we find no inherent boundary conditions but only those determined by proxy, by connecting metric g_oo to Newton's potential by ad hoc condition g_oo which tells us that g_oo as .⁷ Indeed, it has been shown that any change of metrics is possible which produce g_oo as and, therefore, an infinite number of solutions are possible. For example, in addition to condition g_oo(Schwarzchild solution), we also have conditions g_oo (Brillouin solution) and g_oo (Fock solution).⁸ Each of these solutions gives the same metric boundary condition g_oo as . Clearly, Einstein's time-time metric g_oo cannot be uniquely determined because its boundary conditions cannot be inherently defined.

The Speed of Gravity

Finally, condition g_oo which appears in equations (5) and (6) is a contradictory equality. Newton's gravitational potential implies Newton's force of gravitation which acts instantaneously.⁹ In contrast, Einstein's weak field "gravitational potential" g_oo is claimed to represent not a force but Riemann's curved space which acts by somehow emitting energy waves at the speed of light, a curious claim never confirmed by experiment. In physics, the speed of a wave involves the propagation of energy. Since gravity does not involve the propagation of energy its wave equation is space-like as Laplace and Poisson wave equations of Newton's potential which acts instantaneously.

It is clear that Einstein's theory of gravitation cannot be reconciled with Newton's theory of gravitation.

¹ Einstein after Seven Decades http://einstein.stanford.edu/gen_int/story_of_gpb/gpbsty2.html

² Steven Weinberg, Gravitation and Cosmology, John Wiley & Sons, 1972 pages 79-85 (redshift), pages 188-194 (deflection of light), pages 194-201 (precession of perihelia), pages 201-207 (radar echo delay); Leon Brillouin, Relativity Reexamined, Academic Press 1970 pages 54-55 (redshift), pages 98-99 (deflection of light), page 99 (advance of perihelion).

³ Weinberg above pages 77-79 Newton's Limits equations (3.4.2), (3.4.3), (3.4.5)

⁴ Id Derivation of Einstein's equations pages 152-155 equation (7.1.3)

⁵ Brillouin above page 50.

⁶ A.N. Tikhonov and A.A.Samarski, Equations of Mathematical Physics, Dover Publications 1963 page 301 Chapter V Equations of Elliptic Type and page 191 Chapter 3 Physical problems leading to equations of parabolic type. Formulation of boundary-value problems; W.J. Sternberg and T.L.Smith, The Theory of Potential and Spherical Harmonics, University of Toronto Press, 1944 page 180 Chapter VII The Boundary Value Problems of Potential Theory.

⁷ Weinberg above page 77 equation (3.4.5)

⁸ Brillouin above pages 52-53; Weinberg above pages 179-185.

⁹ Tom Van Flandern, The Speed of Gravity - What the Experiments Say http://www.metaresearch.org/cosmology/speed_of_gravity.asp

Copyright © 2000 by James Constant

By the same author: http://www.coolissues.com/gravitation/sameauthor.htm