The real reason why Einstein introduced the Cosmological Constant

We have always been taught that Einstein introduced the cosmological constant in 1917 in his field equations of General Relativity to satisfy his desire of a static universe. But this is not true, or at least it’s highly misleading, and it hides the true reason why Einstein first thought about the cosmological constant.

Einstein envisioned General Relativity to fulfill the General principle of relativity (that the equations describing the laws of physics take the same form in all frames of reference, including non-inertial ones, and that all frames are physically equivalent), the Equivalence Principle, and Mach’s Principle (the metric tensor is completely caused and determined by the stress-energy tensor). The first one was not achieved, and General Relativity features only General Covariance (the equations describing the laws of physics maintain their form under arbitrary coordinate transformations). Kretschmann showed that any theory can be made generally covariant by mathematical tricks, and what matters is which observers the theory says are really equivalent. The second one also faced changes from the original macroscopic equivalence between an accelerating frame and a gravitational field put forward by Einstein in Special Relativity, to a infinitesimal formulation in the final General Relativity theory. The third one, Mach’s principle, which can be reformulated as the inertial reference frames being completely determined via some causal law by the distribution of all matter-energy in the observable universe, is also not achieved. We explained Mach´s principle in our previous post “Mach´s principle: introduction to modified inertia”. The immediate consequences of Mach´s principle are the following:

The inertial mass of a body should increase with the agglomeration of masses in its neighbourhood.
A body in an otherwise empty universe should have no inertia.
A body should experience an acceleration if nearby masses are accelerated, with the accelerating force having the same direction as the acceleration of the masses.
A rotating mass should generate inside it a Coriolis force.

The last two effects occur in General Relativity, making the theory more Machian than Newtonian gravity, but they do not necessarily imply that they are caused by a change of the inertial properties of the body. They can be interpreted as a change of the inertial frame due to the motion of gravitational masses. The first effect will be discussed later. According to Mach’s principle, in the Earth’s frame of reference the plane of a Foucault pendulum must be pulled around by the gravity of the distant stars. Thirring showed early in 1918 that such as effect, known as Lense-Thirring effect, does occur in General Relativity.

Einstein knew that General Relativity, as originally formulated, was in conflict with the second effect, because the empty solution where the stress-energy tensor is zero everywhere is the Minkowski spacetime of Special Relativity, in which test bodies have the usual inertia. He then introduced the cosmological constant term (a locally unobservable energy density of the vacuum or constant of curvature) in his field equations with the hope that that his field equations with the cosmological term would have no solutions for a zero stress-energy tensor (no matter), and that there would be no inertia in the absence of matter, in accordance with Mach’s principle. However, de Sitter found in 1917 a solution for the field equations with the cosmological constant with zero stress-energy tensor for an expanding universe, and Einstein dismissed the cosmological constant as no longer justified.

According to Mach, the relativity principle should be extended to arbitrary rigid transformations (the transformations that preserve the spatial relations between points but change arbitrarily the relation of a given point to space). Thus, overall rigid rotations and translations of a system should be unobservable, which is not the case in Newtonian theory. But Einstein’s General Relativity was also found to be not Machian in this sense, when Gödel formulated in 1949 a rotating universe solution which had observational effects. This doesn’t necessarily mean that the universe is not Machian, but that General Relativity is not, or at least some of its solutions are not.

After these results, only some solutions in General Relativity were considered Machian, depending on boundary conditions on the cosmological solutions of Einstein’s equations. John Archibald Wheeler noted in the 1957 Chapel Hill Conference on the role of gravitation in physics, that the problem of boundary conditions at large in General Relativity was related to Mach’s principle.

On this basis, and with Sciama’s non-relativistic Machian model of inertia, which we explained in our previous post “History of Modified Inertia”, the relativistic Jordan–Brans–Dicke theory was developed in 1961 as a more Machian competing theory to General Relativity, in which a scalar field as an advanced wave integral over all matter was introduced allowing the gravitational constant to be variable with position and time and approximately satisfying Sciama’s relationship. The field equations for the metric had additional terms containing derivatives of the scalar field, and a new field equation for the scalar field was introduced which, to put it simple, stated that the d’Alembertian of the scalar field is proportional to the trace of the stress-energy tensor. This new theory was meant to satisfy the first and second effects presented earlier in the post. The difference in predictions between the Brans-Dicke theory and General Relativity has been locally tested in the solar system near the Sun through the Shapiro time delay effect measured by the Cassini experiment, showing that the Brans-Dicke dimensionless coupling factor of the scalar field to gravity, which differentiates between the theory and General Relativity, must be so high that Brans-Dicke theory is almost indistinguishable from General Relativity. All of these tests have been performed in the high-acceleration regime of the solar system.

But there is a regime in which General Relativity has not been verified, the low acceleration regime, outside any star system and far away from galactic centers. It is reasonable to think that boundary conditions a la Mach play a more important role in the dynamics here than near large masses in the high acceleration regime. And it is exactly what Machian MOND suggests. We explained MOND in a previous post, and we were the first ones to show that MOND can be reformulated in terms of Mach´s principle. According to the Machian MOND interpolating function as a correction to Newtonian inertia, if the field intensity (the gravitational field intensity without the gravitational constant) of the rest of the universe is greater than that from the local system (in the so called deep MOND regime, such as the flat asymptotic velocity part of the disk in galaxies), it seems that acceleration is defined with respect to the background frame of the universe, and the field intensity of the universe at large must be taken into account non-linearly together with that of local masses to explain the dynamics of the local system. This reasoning, which explains the dark matter effect on galaxy rotation curves in a powerful simple way without free parameters, suggests that nearby masses, which result in high field intensities, make the local dynamics less dependent on boundary conditions, and less Machian.

Could the appropriate boundary conditions explain Machian MOND? Is there a way to introduce these boundary conditions in General Relativity to reproduce Machian MOND’s effective correction to solve galaxy rotation dynamics?

References:

Albert Einstein, Cosmological considerations in the general theory of relativity, 1917

Albert Einstein, On the Foundations of the General Theory of Relativity, 1918

Albert Einstein, The Meaning of Relativity, 1922