There was one problem with Newtonian gravitation that was first identified more than 200 years after Newton’s Principia Mathematica, that arises when trying to use Newton’s laws for an infinite, homogenous, and isotropic universe with constant matter density. Hugo Seeliger in 1894 (and Carl Neumann 20 years before) found it and wrote: “Newton’s laws, applied to the immeasurably extended universe, leads to difficulties and irresolvable contradictions if one regards the matter distributed through the universe as infinitely great”. But what was exactly the problem?
An oversimplified way of understanding the Seeliger-Neumann paradox consists on realizing that for any given force of arbitrary magnitude and direction on a test particle, one can always draw a sphere of source masses that exert that force on the test particle and draw the remaining source masses of the universe in concentric spherical shells that do not exert any gravitational force on the test mass by Newton’s shell theorem (a spherical shell of source masses exerts no net force on a test mass located at any position within the shell). The final picture is an infinite homogeneous and isotropic universe with constant mass density, in which the force on the test particle is undetermined. You may think that the force must be balanced out and zero for symmetry reasons (and probably Newton thought the same way), but if you follow Newtonian gravity, it’s not zero or infinite, is mathematically undefined. More precisely, the gravitational force on a test body is the sum of forces exerted by all other masses, and the integration over all masses to compute the force fails to converge, and Newton cannot tell you the magnitude or direction of the force. And from that force, you should be able to compute an observable (acceleration) of the particle. If you want a more formal description of the paradox, I will leave John Norton’s intuitive explanation and review of Seeliger’s formulation in the description.
Notice that this problem is not meant to prove that Newtonian laws are incompatible with our universe (which is not static with constant density and not perfectly homogeneous, and its observable and causally connected part is not infinite), but to prove an incompatibility with these reasonable assumptions of a static, homogeneous, isotropic universe (which also coincided with the idea of how the universe was at the time of Seeliger).
Among many attempted solutions from different authors, such as Seeliger suggestion of adding an exponential attenuation factor to the inverse square law of gravity, Einstein (agreeing with Seeliger in the problem) also offered a well-known modification to Newton’s law in 1917. He first pointed out that Newton’s laws require a constant limit for the potential at spatial infinity, and this is interpreted as the matter density becoming zero at infinity. The same happens in General Relativity, where local solutions (such as the Schwarzschild metric) are asymptotically Minkowskian and flat. But for an infinite universe, matter density is constant at infinity, and the potential is infinite. Then, he introduced the cosmological constant in the Poisson equation representing this average matter density, and equivalently, the same mechanism in General Relativity. But why did he introduce it in General Relativity? As we explained in one of our previous posts, that was is not because he wanted a static universe, but because he wanted a closed one to satisfy Mach’s principle. We explained Mach’s principle in our previous post. Einstein’s definition of Mach’s principle was that “the metric tensor is completely determined by the stress-energy tensor”, and that in a consistent theory of relativity, there can be no inertia relative to space, but only inertia relative to other bodies (in an empty universe, there should be no inertia). To satisfy the latter, Einstein realized that he needed some boundary conditions in the metric different than flat Minkowskian spacetime. In particular, that at infinity, the time component of the metric increases to infinity while the space component vanishes. This implies that the potential energy becomes infinite at infinity. He tried to introduce these asymptotic conditions in General Relativity, but found that there was no stress-energy tensor satisfying them and consistent with the fixed stars having small velocities with respect to the speed of light. An infinite potential at infinity made no sense to Einstein, because for a bound system according to the virial theorem, high gravitational potentials imply high kinetic energies, and these imply large velocities, in this case, of the stars, which Einstein thought they were more or less “fixed” in the celestial background. Thus, he rejected these boundary conditions and divergence of the metric because the gravitational potentials should be small according to the small velocities of stars. And by introducing the cosmological constant, he achieved a closed universe without the need for specifying boundary conditions.
But today we know that stars very far away do have a very high velocity with respect to us, if we interpret their redshift as coming from their velocity away from us. So perhaps Einstein was wrong in rejecting the divergence of the time component and vanishing of the spatial component at infinity for a local solution like the Schwarzschild metric. We explored the idea of whether the expansion or curvature of the universe at large can explain galaxy rotation curves in General Relativity, but we didn’t discuss a modification of the divergence of the metric at infinity.
Curiously, the theory that explains galaxy rotation curves without the need for dark matter, also exhibits an infinite potential at infinity for its local solution. We explained MOND in our previous posts, and how impressive it is that a simple modification to Newton’s laws with a single new constant can generally remove the need for dark matter in galaxies. MOND’s potential diverges logarithmically at infinity, and exhibits some sort of scale invariance, as the velocities do not depend on the distance from the source. We have also insisted on the reasons why MOND might have an origin in Mach’s principle and the relativity of inertia. Does MOND come from the relativity of inertia and its divergence of the metric at infinity as Einstein initially considered?
Moreover, Norton seems satisfied with a solution to Seeliger’s paradox within Newtonian gravity based on the relativity of acceleration in cosmology, found by Malament in 1995. Thanks to the infinite, homogeneous and isotropic background, the Newtonian accelerations (which are the true observable thing, and not the forces) avoid the issue of the paradox if the following modification to Newtonian kinematics is introduced: The decomposition of gravitational free fall into an inertial trajectory and a gravitational deflection is conventional; we are free to divide free fall motion into any combination of inertial motion and gravitational deflection we please, as long as the latter corresponds to a gravitational potential satisfying Poisson’s equation. This is of course related to Mach’s principle and the relativity of inertia, but Norton doesn’t mention it. We will leave in the references the paper from Norton explaining this solution.
References:
John Norton, A Paradox in Newtonian Gravitation Theory
John Norton, The Cosmological Woes of Newtonian Gravitation Theory
John Norton, The Force of Newtonian Cosmology: Acceleration is Relative
Albert Einstein, Cosmological Considerations in the General Theory of Relativity, 1917