Question

I am a bit confused as to how the Equivalence Principle implies a curved spacetime. Or if it doesn’t imply a curved spacetime, then what exactly makes it necessary to have a curved space time?

I could very well have local inertial frames in flat spacetime in arbitrary coordinates. Particles on which only gravity acts could follow curved paths in flat spacetime and their trajectories would be straight locally.

So what exactly forces the manifold to be curved? How exactly is the result that our spacetime should be curved reached?

Why not just study gravity as a background force field on a flat space time?

Answer 1

The Equivalence Principle does imply a curved spacetime, and it is a cornerstone in the development of General Relativity. In the framework of General Relativity, the presence of mass and energy curves spacetime, and objects move along the curves created by this curvature, giving rise to what we perceive as gravitational attraction.

Step-by-step explanation:

The Equivalence Principle, a key concept in General Relativity, states that locally, the effects of gravity are indistinguishable from acceleration. When applying this principle to a small region of spacetime, it leads to the conclusion that gravity is not merely a force acting in flat space but rather a manifestation of curved spacetime caused by the presence of mass and energy.

The mathematical formalism of General Relativity, expressed through Einstein's field equations, describes how matter and energy influence the curvature of spacetime. In these equations, the curvature (represented by the metric tensor) is directly related to the distribution of mass and energy in the vicinity.

The notion of studying gravity as a background force field on flat spacetime doesn't align with the predictions and observations made possible by General Relativity. The theory has been tested and validated in numerous experiments, including the precise prediction of the bending of light around massive objects (gravitational lensing) and the accurate prediction of the precession of Mercury's orbit.

These successes confirm the necessity of curved spacetime to accurately describe the behavior of gravity on cosmic scales. Therefore, General Relativity's curved spacetime framework provides a more accurate and comprehensive understanding of gravity compared to a flat spacetime background.