Transverse mass

In one of his three Annus mirabilis papers of 1905, on special relativity, Albert Einstein noted that, given a specific definition of the word "force" (a definition which he later agreed was not advantageous), and if we choose to maintain (by convention) Newton's second law of motion F = ma (mass times acceleration equals force), then one arrives at ${\displaystyle \scriptstyle m/(1-v^{2}/c^{2})}$ as the expression for the transverse mass of a fast moving particle. This differs from the accepted expression today, because, as noted in the footnotes to Einstein's paper added in the 1913 reprint, "it is more to the point to define force in such a way that the laws of energy and momentum assume the simplest form", as was done, for example, by Max Planck in 1906, who gave the now familiar expression ${\displaystyle \scriptstyle m/{\sqrt {1-v^{2}/c^{2}}}}$ for the transverse mass.

As historian Arthur I. Miller points out, this is equivalent to the transverse mass predictions of both Einstein and Hendrik Lorentz. Einstein had commented already in the 1905 paper that "With a different definition of force and acceleration, we should naturally obtain other expressions for the masses. This shows that in comparing different theories... we must proceed very cautiously."^[1]

Minkowski's work

Einstein did not immediately appreciate the value of Hermann Minkowski's four-dimensional formulation of special relativity, although within a few years he had adopted it within his theory of gravitation.

Black holes

Einstein denied several times that black holes could form. In 1939 he published a paper that argues that a star collapsing would spin faster and faster, spinning at the speed of light with infinite energy well before the point where it is about to collapse into a Schwarzchild singularity, or black hole.

The essential result of this investigation is a clear understanding as to why the "Schwarzschild singularities" do not exist in physical reality. Although the theory given here treats only clusters whose particles move along circular paths it does not seem to be subject to reasonable doubt that more general cases will have analogous results. The "Schwarzschild cannot be concentrated arbitrarily. And this is due to the fact that otherwise the constituting particles would reach the velocity of light.^[2]

Einstein's argument itself only shows that stable spinning objects have to spin faster and faster to stay stable before the point where they collapse. But it is well understood today (and was understood well by some even then) that collapse cannot happen through stationary states the way Einstein imagined. Nevertheless, the extent to which the models of black holes in classical general relativity correspond to physical reality remains unclear, and in particular the implications of the central singularity implicit in these models are still not understood.

Closely related to his rejection of black holes, Einstein believed that the exclusion of singularities might restrict the class of solutions of the field equations so as to force solutions compatible with quantum mechanics, but no such theory has ever been found.

Cosmological constant

Einstein himself considered the introduction of the cosmological constant in his 1917 paper founding cosmology as a "blunder".^[3] The theory of general relativity predicted an expanding or contracting universe, but Einstein wanted a static universe which is an unchanging three-dimensional sphere, like the surface of a three-dimensional ball in four dimensions.

He wanted this for philosophical reasons, so as to incorporate Mach's principle in a reasonable way. He stabilised his solution by introducing a cosmological constant, and when the universe was shown to be expanding, he retracted the constant as a blunder. This is not really much of a blunder – the cosmological constant is necessary within general relativity as it is currently understood, and it is widely believed to have a nonzero value today.

Heisenberg's work

Finding it too formal, Einstein believed that Werner Heisenberg's matrix mechanics was incorrect. He changed his mind when Erwin Schrödinger and others demonstrated that the formulation in terms of the Schrödinger equation, based on wave–particle duality was equivalent to Heisenberg's matrices.

Einstein also tried to show that Heisenberg's uncertainty principle was not valid. By 1927 he had become convinced of its utility, but he always opposed it.

EPR paradox

In the EPR paper, Einstein, Boris Podolsky and Nathan Rosen argued that quantum mechanics cannot be a complete realistic and local representation of phenomena, given specific definitions of "realism", "locality", and "completeness". The modern consensus is that Einstein's concept of realism is too restrictive.