Einstein's general theory of relativity can indeed be broken down into two fundamental aspects: (1) the equation of motion in curved spacetime, and (2) how spacetime is curved in the presence of energy/matter. However, these two aspects are not entirely independent but intricately interconnected within the theory.
Let's discuss each part separately:
- Equation of Motion in Curved Spacetime: The equation of motion in general relativity is described by the Einstein field equations. These equations relate the curvature of spacetime to the distribution of energy and matter within it. They can be written as:
R_{μν} - (1/2)g_{μν}R = (8πG/c^4)T_{μν}
Here, R_{μν} represents the Ricci curvature tensor, R is the scalar curvature, g_{μν} denotes the metric tensor that characterizes the geometry of spacetime, T_{μν} represents the stress-energy tensor (describing the energy and momentum content of matter and fields), G is the gravitational constant, and c is the speed of light in a vacuum.
These equations govern the behavior of matter and energy in the presence of gravity, determining how objects move and interact in a curved spacetime. The equation of motion arises from the curvature of spacetime itself, and it is influenced by the distribution of energy and matter.
- Curvature of Spacetime in the Presence of Energy/Matter: The curvature of spacetime is determined by the distribution of energy and matter within it. This distribution creates a gravitational field that curves the fabric of spacetime itself. In turn, the curved spacetime influences the motion of objects within it, as described by the equation of motion mentioned above.
The presence of energy and matter generates what is known as the stress-energy tensor in the Einstein field equations. This tensor encapsulates the distribution of energy, momentum, pressure, and density of matter and fields. The Einstein field equations then relate this stress-energy tensor to the curvature of spacetime, providing a dynamic and interactive picture of how matter-energy and spacetime geometry are intertwined.
So, while the equation of motion in curved spacetime and the curvature of spacetime in the presence of energy/matter can be conceptually separated, they are intimately connected within the framework of general relativity. The distribution of energy and matter affects the geometry of spacetime, and the resulting curvature, in turn, influences how objects move and interact in that spacetime.