In general relativity, the curvature of space-time caused by massive objects leads to the phenomenon of gravity, which is an attractive force between objects. The reason why massive objects curve space-time in a way that results in attraction rather than repulsion is deeply rooted in the nature of space-time itself and the properties of matter.
According to general relativity, the curvature of space-time is determined by the distribution of matter and energy. Massive objects, such as stars or planets, create a curvature in space-time around them. This curvature influences the motion of other objects in the vicinity, causing them to move along curved paths in response to the gravitational field.
The curvature of space-time can be understood in terms of the relationship between matter and the geometry of space-time. Matter tells space-time how to curve, and curved space-time tells matter how to move. This interplay is governed by the equations of general relativity, which describe how matter and energy interact with the geometry of space-time.
When an object moves in the curved space-time around a massive object, it follows the path of least resistance, which corresponds to the path determined by the curvature of space-time. In the presence of a massive object, the curved space-time creates a "well" or a "dip" in which other objects tend to fall. This falling motion is what we perceive as gravitational attraction.
The reason why the curvature of space-time caused by massive objects leads to attraction rather than repulsion is related to the positive energy density associated with matter. Massive objects have positive energy densities, and according to the equations of general relativity, positive energy densities lead to attractive gravitational effects.
In contrast, repulsive effects in the context of general relativity are associated with negative energy densities or exotic forms of matter, such as dark energy, which has a repulsive gravitational effect on large scales. However, in everyday circumstances involving ordinary matter, the energy densities are positive, leading to attractive gravitational forces.
It's important to note that the explanation above provides a conceptual understanding of why gravity is attractive in general relativity. The precise mathematical formulation of gravity and its attractive nature is described by the equations of general relativity, which involve complex mathematical expressions and tensors.