# Are there good reasons why special relativity should motivate geometrised gravity in a way that Newtonian mechanics does not?

I have studied a bit of Newton Cartan theory recently, the geometrised version of Newtonian gravity in which gravity is due to the curvature of spacetime, but is Newtonian (simultaneity is absolute).

This theory allows us to fill in a table of theories in which we have geometrised vs non-geometrised gravity as the columns, and the rows are absolute time vs constant speed of light. Such a table would look like

$$\begin{array}{cc} \textrm{Newton-Cartan} & \textrm{Newton}\\ \textrm{GR} & \textrm{SR} \end{array}$$

On consideration of such a table, I wondered why I always thought that the motivation for geometrised gravity came from relativity. Clearly they are independent constraints on a theory, since every box of the table is filled.

So, my question is whether there are good reasons that special relativity should motivate geometrised gravity in a way that Newtonian mechanics does not.

• I'm not so sure that your 2x2 table makes sense. Newtonian gravity and Newton-Cartan are two different formalizations of the same theory. They make the same predictions. GR and SR are not the same theory. SR is a special case of GR. – user4552 Jul 23 '19 at 12:26
• It is debated whether Newtonian gravity is equivalent to Newton-Cartan, i.e. do they agree as to what frames are intertial? Nonetheless, SR is a theory without gravity unlike the other three. I guess you could interpret the columns as curved vs flat connection? – Joshua Tilley Jul 24 '19 at 13:58

## 1 Answer

1. Firstly, Newtonian gravity (NG) has a self-consistent non-geometrized formulation, unlike relativistic gravity.

2. Secondly, the relativistic metric is non-degenerate unlike in Newton-Cartan (NC) theory, which introduces 2 degenerate metrics. This makes NC theory technically more challenging and unlikely to be invented before its relativistic counterpart.

• When you say that relativistic gravity has no non-geometric formulations, what do you mean? Would you call gauge theories of gravitation (where space time is flat) "geometric"? – Lorenz Mayer Jul 23 '19 at 8:53
• @LorenzMayer, I would argue that gauge theories of gravitation in flat space-time aren't a true representation of GR and relativistic mechanics. Once you get rid of the influence mass has on spacetime you aren't really working in relativity or Quantum Gravity, just a sort of semi-classical analogue. – huntercallum Jul 24 '19 at 8:10
• it is true that gauge theories are not GR, that's the whole point - they are "non-geometric". This is talking about classical theories. For quantum theories, nobody came up with something convincing, and there is no a priori reason why gauge theories shouldn't work. – Lorenz Mayer Jul 24 '19 at 9:03
• Is anyone aware of a proof of the claim that GR cannot be equivalent to a "non-geometrized" theory? I presume the latter would mean a theory with a flat connection. – Joshua Tilley Jul 24 '19 at 14:00