# General relativity: is curvature of spacetime really required or just a convenient representation?

I'm not really far into the general theory of relativity but already have an important question: are there formulations that can do without spacetime curvature and describe the general theory of relativity/all associated gravitational effects in global cartesian coordinates? Idea: Einstein chose spacetime curvature so that one had not to build gravitational effects into the rest of physics equations like Maxwell's equations. I assume that spacetime curvature provides a convenient way to apply physics laws not related to gravity unmodified locally because at small distances we may use special relativity as an approximation.

Please correct me everywhere I am wrong.

The curvature of spacetime is a property of the spacetime manifold itself, it is not related to any particular choice of coordinates. The very essence of relativity lies in the Einstein field equations, which, colloquially, tell you that the energy-matter content of spacetime determines its curvature and metric. There is no formulation of general relativity known to me that avoids the idea of curved spacetime.

But this is not worrying - generally, such a geometric theory of physics is the kind we want. For classical mechanics, such a geometric picture is given by the Hamiltonian formalism and its symplectic manifolds. For gauge theories describing the other fundamental forces except for gravity, the geometric picture is given by the theory of principal bundles and the (e.g. electromagnetic) physical fields are amenable to be described as the curvature of such bundles (and wonderfully analogous to formulating GR with jet and frame bundles). For gravity, the geometric picture is that it is spacetime itself that has curvature. We don't even want to get rid of a description in terms of curvature, generally speaking.

Just because you can locally choose frames such that, at least at one point, the metric is flat, and hence you have SR, this is different from saying that the curvature is not required - a spacetime is flat only when there is a choice of coordinates such that the metric is flat everywhere.

• I am reading your answer to is curvature of spacetime really required ?. You say, "We don't even want to get rid of a description in terms of curvature". Why won't we? Why not "imbed" this curved space in a flat space of higher dimension, and instead of curvature (that we'll have further to explain why is the space curved), we can speak of masses and forces of attraction between them? It's more materialistic. Commented Jan 13, 2015 at 21:55
• @Sofia: Embedding curved spaces in flat space doesn't get rid of the fact that they are curved (a sphere is still curved after embedding it in $\mathbb{R}^3$). And I say that "we" don't want to get rid of the curvature because geometric theories like gauge theories and GR perfectly show how "forces" arise from an action principle - which is aesthetically satisfying and amenable to quantization. I see no advantage in trying to get away from geometric theories, since they are a success story (e.g., I think of gauge theories here, and how they solved the confusion that was the particle zoo). Commented Jan 14, 2015 at 17:32

There are formulations of classical general relativity that do not utilize the concept of the curvature of spacetime, which are equivalent to the traditional formulation that interprets gravity as the curvature of spacetime.

As with anything else, the framework that one chooses to do calculations in is a matter of circumstance, or philosophy, etc... sometimes both. But, history of science has shown time and time again, that it is well worth it to explore alternative formulations of theories that are known to be successful.

Quoted from the abstract of the paper above, "In these notes we discuss two alternative, though equivalent, formulations of General Relativity in flat spacetimes, in which gravity is fully ascribed either to torsion or to non-metricity, thus putting forward the existence of three seemingly unrelated representations of the same underlying theory."

This seems pretty clear that these operate in flat spacetime, i.e. as you say, "global cartesian coordinates." I can write a more thorough explanation if required by the OP.

• These are variants of teleparallel gravity and first considered by Einstein. They use a metric as well as a connection. It's the connection that has torsion and lacks curvature. The underlying metric still remains and has curvature ... Commented Mar 28, 2021 at 3:19
• I never said there was no curvature, but rather that gravity itself is not interpreted as the curvature of the metric (it is in other properties, e.g. torsion and non-metricity. Commented Mar 28, 2021 at 3:27
• I'm not sure that's the best interpretation of it. Look at the Cartan-Einstein theory for a theory of gravity that incorporates torsion. There, there is no torsion outside of the matter fields, that is torsion is a consequence of matter. Commented Mar 28, 2021 at 3:31
• In Einstein-Cartan theory, the torsion is associated with spin, and that theory operates in a different geometry entirely, the Riemann-Cartan geometry, than general relativity, so we can't compare them so casually... Commented Mar 28, 2021 at 3:39
• Actually you can, the tetrad field in teleparallel gravity uses a spin connection. Commented Mar 28, 2021 at 3:42

In traditional GR, the possibility that spacetime can be curved is a fundamental requirement and moreover gravity is a fundamental force. However, in entropic gravity, it's argued that gravity is not fundamental and is an entropic effect and hence curvature is also not fundamental but an emergent property.

• Could you provide a source for your claim that "spacetime curvature is fundamental requirement and that gravity is a fundamental force"? In classical GR, it is up to philosophical interpretation as to whether or not the connection or metric is "fundamental." And in GR there is no concept as "fundamental force." It seems you're mixing concepts here. Commented Mar 28, 2021 at 13:18
• Not quite: Have a look at the comment that I've added to your answer. Commented Mar 31, 2021 at 0:15
• I have... I've responded to your comments there. If you cant provide sources then this is kind of a waste of time. Commented Apr 3, 2021 at 12:19
• @Daddy Kropotkin: I've given you a reference where it clearly said that teleparallel fravity was equivelent to GR only locally and then only without spinless matter. Since matter is fundamental to gravity, and matter is fermionic, and hence spinning, we see that teleparallel gravity is not exactly equivalent to GR - at all. You're making spurious claims. It's you that is wasting my time. Commented Apr 3, 2021 at 12:22
• Wow you're getting defensive now.... and I read that paper you cited, and I also quoted from it where it says that the theory known as Torsion Gravity is equivalent to GR and does not have curvature aspect of gravity. I am not making spurious claims. Nice try. Commented Apr 3, 2021 at 13:59