In high school, I completed AP Physics C Mechanics and Electricity and Magnetism. In my first year of undergrad, I completed a course on Newtonian Mechanics and a course on Special Relativity and Electromagnetism which both approximately followed the sections on those topics in the Feynman Lectures on Physics.
In my free time, I am starting to learn tensor analysis and general relativity. I wanted to explain what my current understanding of GR is and was wondering if what I understand of it so far could be verified and if it is not correct the problems with it could be explained.
My Current Understanding:
- Objects follow geodesic in spacetime which extremize the total spacetime distance (proper time) along that geodesic. These geodesics can be found through the geodesic equation if you know the metric tensor.
- The energy-momentum tensor measures how much energy density/flow, momentum density/flow is in a certain region of spacetime.
- The energy-momentum tensor determines what the metric tensor is through the Einstein field equation.
- If the energy-momentum tensor is known, the Einstein field equation can be used to solve for the metric tensor (i.e. the Schwarzschild metric is the solution for the metric tensor if the energy-momentum tensor is that of a spherical star or black hole). Then the geodesic equation can be used to calculate the trajectory of any object in spacetime.
To summarize, the energy/mass existing at a point in spacetime causes the spacetime around it to curve and this curvature influences the movement of objects that travel through the "shortest path" through spacetime.
- Does the energy-momentum tensor vary with the spacetime coordinates just like the metric tensor does and is it determined by the distribution of energy and momentum throughout spacetime (i.e. if a massive body exists somewhere)?
- If that is so, does the value of the energy-momentum tensor at a point in spacetime influence the curvature of spacetime only at that specific point or does it influence the curvature of surrounding points in spacetime as well (i.e. does the Sun cause spacetime to curve in a large region around it or just at the points in spacetime where the Sun exists)?