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When the conservation of the stress–energy tensor (=) for a perfect fluid is combined with the conservation of particle number density (=), both utilizing the 4-gradient, one can derive the relativistic Euler equations, which in fluid mechanics and astrophysics are a generalization of the Euler equations that account for the effects of ...
In general relativity, one usually refers to "the" covariant derivative, which is the one associated with Levi-Civita affine connection. By definition, Levi-Civita connection preserves the metric under parallel transport, therefore, the covariant derivative gives zero when acting on a metric tensor (as well as its inverse).
Given the covariant derivative, one can define the parallel transport of a vector v at a point P along a curve γ starting at P. For each point x of γ, the parallel transport of v at x will be a function of x, and can be written as v(x), where v(0) = v. The function v is determined by the requirement that the covariant derivative of v(x) along ...
In the fundamental branches of modern physics, namely general relativity and its widely applicable subset special relativity, as well as relativistic quantum mechanics and relativistic quantum field theory, the Lorentz transformation is the transformation rule under which all four-vectors and tensors containing physical quantities transform from one frame of reference to another.
The relativistic four-velocity, that is the four-vector representing velocity in relativity, is defined as follows: = = (,) In the above, is the proper time of the path through spacetime, called the world-line, followed by the object velocity the above represents, and
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 principle, the derivative of a function can be computed from the definition by considering the difference quotient and computing its limit. Once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones.
Following the non-relativistic approach, we expect the derivative of this seemingly correct Lagrangian with respect to the velocity to be the relativistic momentum, which it is not. The definition of a generalized momentum can be retained, and the advantageous connection between cyclic coordinates and conserved quantities will continue to apply ...