The relationship between Kaluza-Klein theory and general relativity is a foundational aspect of the former. Kaluza-Klein theory extends general relativity by incorporating additional dimensions beyond the familiar four-dimensional spacetime. Here are some key points from the document regarding this relationship:

  1. Extension of General Relativity: Kaluza-Klein theory applies Einstein’s general theory of relativity to a five-dimensional spacetime manifold instead of the usual four dimensions. This is done without introducing a five-dimensional energy-momentum tensor, implying that the universe in higher dimensions is considered empty. This idea aligns with Kaluza’s first key assumption: to explain matter in four dimensions as a manifestation of pure geometry in higher dimensions​​.
  2. Generalized Einstein Equations: In the Kaluza-Klein framework, the Einstein equations are generalized to five dimensions. The equations G^AB=0Ĝ_{AB} = 0 or R^AB=0R̂_{AB} = 0, where G^ABĜ_{AB} and R^ABR̂_{AB} are the five-dimensional Einstein and Ricci tensors, respectively, reflect this higher-dimensional perspective【12:​​. Minimal Extension and Physical Interpretation: The five-dimensional Ricci tensor and Christoffel symbols are defined similarly to their four-dimensional counterparts, which can lead to significant departures from general relativity. However, these departures have not been widely observed in typical tests conducted within the solar system. The sun, for instance, is very close to the Schwarzschild limit, leading to minimal deviations from general relativity in observed solar phenomena【12:​​. Noncompactified Kaluza-Klein Theory and Astrophysical Implications: Noncompactified Kaluza-Klein theory, where higher-dimensional dependencies are allowed, offers new insights into cosmology and astrophysics. For example, in the context of noncompactified Kaluza-Klein cosmology, phenomena like the big bang, Hubble expansion, and microwave background can be reinterpreted as geometrical illusions—artifacts of coordinate choices in the higher-dimensional universe【12:​​n summary, Kaluza-Klein theory builds upon and extends general relativity by incorporating additional spatial dimensions, leading to a unified framework that can describe both gravitational and electromagnetic forces within a higher-dimensional spacetime. This relationship allows for a richer theoretical structure that provides new perspectives on cosmological and astrophysical phenomena.