There are alternate ways of writing four-vector expressions in physics: The Latin tensor index ranges in {1, 2, 3}, and represents a 3-space vector, e.g.
The tensor contraction used in the Minkowski metric can go to either side (see Einstein notation):[1]: 56, 151–152, 158–161
The 4-gradient covariant components compactly written in four-vector and Ricci calculus notation are:[2][3]: 16
The strong equivalence principle can be stated as:[4]: 184 "Any physical law which can be expressed in tensor notation in SR has exactly the same form in a locally inertial frame of a curved spacetime."
The 4-gradient commas (,) in SR are simply changed to covariant derivative semi-colons (;) in GR, with the connection between the two using Christoffel symbols.
The 4-gradient is used in a number of different ways in special relativity (SR): Throughout this article the formulas are all correct for the flat spacetime Minkowski coordinates of SR, but have to be modified for the more general curved space coordinates of general relativity (GR).
It gets canceled when taking the 4D dot product since the Minkowski Metric is Diagonal[+1,−1,−1,−1].
representing gravitational radiation in the weak-field limit (i.e. freely propagating far from the source).
is the equivalent of a conservation equation for freely propagating gravitational waves.
is a Lorentz scalar invariant shows that the total derivative with respect to proper time
is a mathematical object that describes the electromagnetic field in spacetime of a physical system.
The operator is named after French mathematician and physicist Jean le Rond d'Alembert.
Some examples of the 4-gradient as used in the d'Alembertian follow: In the Klein–Gordon relativistic quantum wave equation for spin-0 particles (ex.
is the transverse traceless 2-tensor representing gravitational radiation in the weak-field limit (i.e. freely propagating far from the source).
More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface.
The HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave.
In this sense, the HJE fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the 18th century) of finding an analogy between the propagation of light and the motion of a particle The generalized relativistic momentum
of the system; a test particle in a field using the minimal coupling rule.
The relativistic Hamilton–Jacobi equation is obtained by setting the total momentum equal to the negative 4-gradient of the action
The (spatial components) de Broglie matter wave relation
It is nice that the gamma matrices themselves refer back to the fundamental aspect of SR, the Minkowski metric:[7]: 130
[3][7] Start with the standard SR 4-vectors:[1] Note the following simple relations from the previous sections, where each 4-vector is related to another by a Lorentz scalar: Now, just apply the standard Lorentz scalar product rule to each one:
If the rest mass term is set to zero (light-like particles), then this gives the free Maxwell equation:
More complicated forms and interactions can be derived by using the minimal coupling rule: In modern elementary particle physics, one can define a gauge covariant derivative which utilizes the extra RQM fields (internal particle spaces) now known to exist.
The full covariant derivative for the fundamental interactions of the Standard Model that we are presently aware of (in natural units) is:[3]: 35–53
) here refer to the internal spaces, not the tensor indices: The coupling constants
However, a line integral involves the application of the vector dot product, and when this is extended to 4-dimensional spacetime, a change of sign is introduced to either the spatial co-ordinates or the time co-ordinate depending on the convention used.
In this article, we place a negative sign on the spatial coordinates (the time-positive metric convention
The factor of (1/c) is to keep the correct unit dimensionality, [length]−1, for all components of the 4-vector and the (−1) is to keep the 4-gradient Lorentz covariant.
Regarding the use of scalars, 4-vectors and tensors in physics, various authors use slightly different notations for the same equations.