# Why is the principle of equivalence necessary as a foundation in GTR?

• A
• e2m2a

#### e2m2a

TL;DR Summary
What is the relevance of the principle of equivalence in the general theory of relativity. Constant accelerating reference frames do not generate "tidal" forces in their frames.
I am studying the general theory of relativity(GTR). Covariance and the principle of equivalence are foundational pillars for the theory. I can understand the need for covariance but I don't see why the need for the principle of equivalence (POE). What I have seen so far is that the properties and curvature of spacetime due to mass/energy via the energy-momentum tensor(EMT) must be described by using the Riemann tensor, Ricci tensor, and the Ricci scalar, among other mathematical objects such as the Christoffel symbols. But these curvature measuring tensors and non-tensors are not applicable in an accelerating reference frame because real curvature does not occur in accelerating systems that are accelerating in flat spacetime. So why did EInstein cite the POE as a necessary foundation? Are Christoffel symbols even relevant in linearly accelerating reference frames? I understand that POE is defined for a homogenous gravitational field locally. But globally the POE does not admit tidal forces. So why even the need for the POE as a conceptual footing for GTR?

Last edited:

Summary:: What is the relevance of the principle of equivalence in the general theory of relativity. Constant accelerating reference frames do not generate "tidal" forces in their frames.

Are Christoffel symbols even relevant in linearly accelerating reference frames?
Yes. Very very very relevant.

Yes. Very very very relevant.
How are Christoffel symbols used in linearly accelerating systems?

How are Christoffel symbols used in linearly accelerating systems?
Christoffel symbols are relevant to any curvilinear coordinate system (as it is in Euclidean space as well), which is effectively what an accelerated coordinate system in Minkowski space is by definition.

How would the theory look if gravitational mass was not equal to inertial mass?

How would the theory look if gravitational mass was not equal to inertial mass?
It would not be a geometric theory.

• Dale
How would the theory look if gravitational mass was not equal to inertial mass?
mmm. let me think about that.

globally the POE does not admit tidal forces
There is no "globally" for the POE. It is only a local principle. That's the whole point: every curved spacetime looks locally like flat spacetime, but only locally.

Summary:: What is the relevance of the principle of equivalence in the general theory of relativity. Constant accelerating reference frames do not generate "tidal" forces in their frames.

. I can understand the need for covariance but I don't see why the need for the principle of equivalence (POE).
The POE basically says that gravity can be described as spacetime geometry. So it is pretty central to GR. Any theory that respects the POE can be geometrized and any geometrical theory of gravity respects the POE.

• vanhees71, Ibix and PeterDonis
An alternative is to make Poincare symmetry of SR local, which also leads to a spacetime geometrical description of the gravitational interaction, and the EP is derived in this way. In the most general case when matter with non-zero-spin particles is present, it's however an extension of GR to Einstein-Cartan theory (i.e., a differentiable manifold with pseudometric of signature (1,3) or (3,1) with a metric compatible connection and torsion).

• Dale