Fundamentals of General Relativity

• physical system: a physical system consists of spacetime itself, along with the matter and energy distributed within it. Spacetime is not a static background but a dynamic entity that interacts with matter and energy.
  • physical systems in GR:
    • single star or planet (e.g., the Sun or Earth) curving spacetime around it.
    • black hole, which is a region of spacetime with extreme curvature
    • the entire universe, described by cosmological models (e.g., the Friedmann-Lemaître-Robertson-Walker metric).
  • The physical system is characterized by the metric tensor $g_{\mu\nu}$, which describes the geometry of spacetime, and the stress-energy tensor $T_{\mu\nu}$, which describes the distribution of matter and energy.
• state of the physical system: state of a physical system in GR is described by the geometry of spacetime (encoded in the metric tensor $g_{\mu\nu}$) and the distribution of matter and energy (encoded in the stress-energy tensor $T_{\mu\nu}$).
  • The state of the system is determined by solving the Einstein field equations, which relate the curvature of spacetime to the distribution of matter and energy: \(G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}\), where $G_{\mu\nu}$ is the Einstein tensor (encoding the curvature of spacetime), $G$ is the gravitational constant, and $c$ is the speed of light.
  • states in GR:
    • Schwarzschild solution, describing the spacetime around a non-rotating, spherically symmetric mass (e.g., a black hole or star
    • Kerr solution, describing the spacetime around a rotating black hole.
    • Friedmann-Lemaître-Robertson-Walker (FLRW) metric, describing the expanding universe in cosmology.
• physical law: the fundamental principles and equations that govern the behavior of spacetime and its interaction with matter and energy. These laws are derived from the Einstein field equations and the principles of general covariance and equivalence.
  • physical laws in GR include:
    • Einstein field equation
    • geodesic equation: describes the motion of particles in curved spacetime
    • Bianchi identities: ensure the consistency of the Einstein field equations
    • principle of general covariance: states that the laws of physics must be expressed in a form that is independent of the coordinate system.
    • equivalence principle: states that the effects of gravity are locally indistinguishable from acceleration.
  • Physical laws in GR are subject to conditions of applicability. For example:
    • GR is valid in regimes where gravitational fields are strong and spacetime curvature is significant.
    • it reduces to Newtonian gravity in the weak-field, low-velocity limit.