In general relativity, the theory developed by Albert Einstein, the mechanism by which matter affects the geometry of space is described by the Einstein field equations. These equations establish a relationship between the distribution of matter and energy in the universe and the curvature of spacetime.
According to general relativity, the presence of matter and energy curves the fabric of spacetime itself. This curvature is what we perceive as the gravitational force. In other words, massive objects such as stars, planets, or any form of energy create a gravitational field that curves the surrounding spacetime.
The mechanism behind this curvature can be understood through the concept of stress-energy. Matter and energy contribute to the stress-energy tensor, which describes the distribution of mass, momentum, pressure, and energy density in spacetime. The stress-energy tensor is used in the Einstein field equations to determine the curvature of spacetime.
The Einstein field equations are a set of differential equations that relate the curvature of spacetime (described by the metric tensor) to the distribution of matter and energy. By solving these equations, one can determine the specific geometry of spacetime in the presence of different matter and energy configurations.
It's important to note that the mechanism by which matter affects the geometry of space in general relativity is a purely mathematical description. The theory does not provide a microscopic explanation of how gravity emerges from the fundamental interactions between particles. This is an area of active research in physics, and various hypotheses and theories, such as string theory or quantum gravity, aim to address this question and reconcile general relativity with quantum mechanics on a fundamental level.