Lorentz ether theory

Hendrik Lorentz and Henri Poincaré developed their version of special relativity in a series of papers from about 1900 to 1905. They used Maxwell's equations and the principle of relativity to deduce a theory that is mathematically equivalent to the theory later developed by Einstein.

Taiji relativity

Taiji relativity is a formulation of special relativity developed by Jong-Ping Hsu and Leonardo Hsu.^{[1][11][12][13]} The name of the theory, Taiji, is a Chinese word which refers to ultimate principles which predate the existence of the world. Hsu and Hsu claimed that measuring time in units of distance allowed them to develop a theory of relativity without using the second postulate in their derivation.

It is the principle of relativity, that Hsu & Hsu say, when applied to 4d spacetime, implies the invariance of the 4d-spacetime interval ${\displaystyle s^{2}=w^{2}-r^{2}}$. The difference between this and the spacetime interval ${\displaystyle s^{2}=c^{2}t^{2}-r^{2}}$ in Minkowski space is that ${\displaystyle s^{2}=w^{2}-r^{2}}$ is invariant purely by the principle of relativity whereas ${\displaystyle s^{2}=c^{2}t^{2}-r^{2}}$ requires both postulates. The "principle of relativity" in spacetime is taken to mean invariance of laws under 4-dimensional transformations. They show that there are versions of relativity which are consistent with experiment but have a definition of time where the "speed" of light is not constant. They develop one such version called common relativity which is more convenient for performing calculations for "relativistic many body problems" than using special relativity.

Several authors have made the case that Taiji relativity still assumes a further postulate - the cosmological principle that time and space look the same in all directions.^[14] Behara (2003) wrote that "the postulation on the speed of light in special relativity is an inevitable consequence of the relativity principle taken in conjunction with the idea of the homogeneity and isotropy of space and the homogeneity of time in all inertial frames".^[15]

Minkowski spacetime

Minkowski space (or Minkowski spacetime) is a mathematical setting in which special relativity is conveniently formulated. Minkowski space is named for the German mathematician Hermann Minkowski, who around 1907 realized that the theory of special relativity (previously developed by Poincaré and Einstein) could be elegantly described using a four-dimensional spacetime, which combines the dimension of time with the three dimensions of space.

Mathematically there are a number of ways in which the four-dimensions of Minkowski spacetime are commonly represented: as a four-vector with 4 real coordinates, as a four-vector with 3 real and one complex coordinate, or using tensors.

Spacetime algebra

Spacetime algebra is a type of geometric algebra, closely related to Minkowski space, and is equivalent to other formalisms of special relativity. It uses mathematical objects such as bivectors to replace tensors in traditional formalisms of Minkowski spacetime, leading to much simpler equations than in matrix mechanics or vector calculus.

de Sitter relativity

According to the works of Cacciatori, Gorini, Kamenshchik,^[7] Bacry and Lévi-Leblond^[17]and the references therein, if you take Minkowski's ideas to their logical conclusion, then not only are boosts non-commutative but translations are also non-commutative. This means that the symmetry group of space time is a de Sitter group rather than the Poincaré group. This results in spacetime being slightly curved even in the absence of matter or energy. This residual curvature is caused by a cosmological constant to be determined by observation. Due to the small magnitude of the constant, then special relativity with the Poincaré group is more than accurate enough for all practical purposes, although near the Big Bang and inflation de Sitter relativity may be more useful due to the cosmological constant being larger back then. Note this is not the same thing as solving Einstein's field equations for general relativity to get a de Sitter Universe, rather the de Sitter relativity is about getting a de Sitter Group for special relativity which neglects gravity.

Euclidean relativity

Euclidean relativity^{[18][19][20] [21] [22][23][24]} uses a Euclidean (++++) metric in four-dimensional Euclidean space as opposed to the traditional Minkowski (+---) or (-+++) metric in four-dimensional space-time.^{[lower-alpha 1]} The Euclidean metric is derived from the Minkowski metric by rewriting ${\displaystyle (cd\tau )^{2}=(cdt)^{2}-dx^{2}-dy^{2}-dz^{2}}$ into the equivalent ${\displaystyle (cdt)^{2}=dx^{2}+dy^{2}+dz^{2}+(cd\tau )^{2}}$. The roles of time t and proper time ${\displaystyle \tau }$ have switched so that proper time ${\displaystyle \tau }$ takes the role of the coordinate for the 4th spatial dimension. A universal velocity ${\displaystyle c}$ for all objects moving through four-dimensional space appears from the regular time derivative ${\displaystyle c^{2}=(dx/dt)^{2}+(dy/dt)^{2}+(dz/dt)^{2}+(cd\tau /dt)^{2}}$. The approach differs from the so-called Wick rotation or complex Euclidean relativity. In Wick rotation, time ${\displaystyle t}$ is replaced by ${\displaystyle it}$, which also leads to a positive definite metric, but it maintains proper time ${\displaystyle \tau }$ as the Lorentz invariant value whereas in Euclidean relativity ${\displaystyle \tau }$ becomes a coordinate. Because ${\displaystyle c^{2}=(dx/dt)^{2}+(dy/dt)^{2}+(dz/dt)^{2}+(cd\tau /dt)^{2}}$ implies that photons travel at the speed of light in the subspace {x, y, z} and baryonic matter that is at rest in {x, y, z} travels normal to photons along ${\displaystyle {\tau }}$, a paradox arises on how photons can be propagated in a space-time. The possible existence of parallel space-times or parallel worlds shifted and co-moving along ${\displaystyle \tau }$ is the approach of Giorgio Fontana.^[25] Euclidean geometry is consistent with Minkowski's classical theory of relativity. When the geometric projection of 4D properties to 3D space is made, the hyperbolic Minkowski geometry transforms into a rotation in 4D circular geometry.