The field equation in general relativity is meant to serve as modern physics’ fundamental explanation of gravity; no longer as a force, but as the manifestation of spacetime curvature. Given this conceptual shift, one might expect that the framework would be entirely free of Newtonian mechanics, which it was intended to supersede.
While referencing Newton’s law in the weak-field limit is reasonable for comparison and testing, the explicit presence of Newton’s gravitational constant 𝐺 in the field equation appears conceptually inconsistent. Could someone clarify why this is the case?
Newton's constant is the coupling constant for gravity, it's a measure of how matter is affected by and affects the spacetime geometry. Just like Fermi's constant is one way of describing how matter that carries the weak charge is affected by and affects other carriers of the weak charge.
Dear Stam Nicolis
Unlike the theory of weak interaction, which remains valid within its domain, Newtonian gravity has been largely superseded by general relativity. Yet, it seems that Einstein’s field equations began with the structure of Newtonian gravity and were then adjusted to account for anomalies such as Mercury’s orbital precession. Along the way, powers of 𝑐 appear to have been added primarily to ensure dimensional consistency, rather than emerging from first principles.
Newton's constant may appear in general relativity and its approximation, Newtonian gravity, this doesn't mean, however, that the appearance of Newton's constant in general relativity is the sign of any inconsistency. That statement is wrong. If one sets G=0, one simply considers spacetime geometries in the absence of matter.
The way Einstein discovered his equations doesn't imply anything besides history. Their meaning is invariance under general coordinate transformations. And the only way to couple matter consistently is through its energy-momentum tensor. There's a coupling constant involved. And in the non-relativistic limit one identifies this constant with Newton's constant. It's not a good idea to try to learn physics by reading the historical development.
The velocity of light doesn't just appear for dimensional reasons, there's a more substantial reason, namely it is possible to show that at any given point of spacetime it is possible to define a reference frame invariant under Lorentz transformations. That's how the speed of light appears. If the spacetime is flat, then this frame can be globally defined,if it's curved, it can't. All this is presented in any textbook on general relativity, which it would be a good idea to study. There are many open problems in general relativity, but this isn't one of them. Which constants appear, where they appear and why they appear is fully unddrstood.
Dear Stam Nicolis
Although the gravitational constant G - originally a force constant in Newtonian gravity - is reintroduced in general relativity as a coupling factor between spacetime curvature and the stress-energy tensor, its inclusion raises conceptual questions.
If Newtonian gravity has been fundamentally replaced by general relativity, why does a key element of that classical theory - complete with the same physical dimension - persist in the new framework?
It seems that G was borrowed as a convenient tuning parameter, not derived from deeper relativistic principles. To complete the dimensional consistency, additional factors of the speed of light c were incorporated, making the formulation appear more like a patchwork to match empirical data than a purely theoretical leap. One might ask: was this an elegant unification or simply a clever retrofit?
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