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Before the advent of general relativity, changes in physical processes were generally described by partial derivatives, for example, in describing changes in electromagnetic fields (see Maxwell's equations). Even in special relativity, the partial derivative is still sufficient to describe such changes. However, in general relativity, it is ...
To derive the equations of special relativity, one must start with two other The laws of physics are invariant under transformations between inertial frames. In other words, the laws of physics will be the same whether you are testing them in a frame 'at rest', or a frame moving with a constant velocity relative to the 'rest' frame.
In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating total energy (which is also called relativistic energy) to invariant mass (which is also called rest mass) and momentum. It is the extension of mass–energy equivalence for bodies or systems with non-zero momentum.
The expression appears in several equations in special relativity, and it arises in derivations of the Lorentz transformations. The name originates from its earlier appearance in Lorentzian electrodynamics – named after the Dutch physicist Hendrik Lorentz. [2] It is generally denoted γ (the Greek lowercase letter gamma).
In mathematical physics, spacetime algebra (STA) is the application of Clifford algebra Cl 1,3 (R), or equivalently the geometric algebra G(M 4) to physics. Spacetime algebra provides a "unified, coordinate-free formulation for all of relativistic physics, including the Dirac equation, Maxwell equation and General Relativity" and "reduces the mathematical divide between classical, quantum and ...
These equations, together with the geodesic equation, [8] which dictates how freely falling matter moves through spacetime, form the core of the mathematical formulation of general relativity. The EFE is a tensor equation relating a set of symmetric 4 × 4 tensors. Each tensor has 10 independent components.
The equation for the covariant derivative can be written in terms of Christoffel symbols. The Christoffel symbols find frequent use in Einstein's theory of general relativity , where spacetime is represented by a curved 4-dimensional Lorentz manifold with a Levi-Civita connection .
The relative acceleration A μ of the two objects is defined, roughly, as the second derivative of the separation vector X μ as the objects advance along their respective geodesics. Specifically, A μ is found by taking the directional covariant derivative of X along T twice: