Einstein's gravitational equation, known as the field equation of general relativity, is expressed as follows:
Rμν - 1/2 Rgμν = 8πGTμν
In this equation, Rμν represents the Ricci curvature tensor, R represents the scalar curvature, gμν denotes the metric tensor, G is the gravitational constant, and Tμν represents the stress-energy tensor.
The absence of a separate time dimension in Einstein's gravitational equation is due to the nature of general relativity and the way it describes the curvature of spacetime. In general relativity, spacetime is treated as a unified four-dimensional manifold where the curvature is determined by the distribution of matter and energy. It is the interplay between matter/energy and the geometry of spacetime that gives rise to gravity.
Instead of treating time as a distinct dimension, general relativity incorporates time as part of the overall spacetime geometry. In this framework, the metric tensor, gμν, describes the geometry of spacetime, including its temporal aspects. The components of the metric tensor relate to both spatial and temporal intervals, essentially combining the three spatial dimensions with the temporal dimension into a unified whole.
Therefore, in Einstein's gravitational equation, time is not treated as a separate dimension but is incorporated into the overall geometry of spacetime. This approach allows general relativity to provide a comprehensive description of gravity, which is not limited to the effects of purely spatial dimensions but also encompasses the influence of time.