Second postulate (invariance of c) The two-postulate basis for special relativity is the one historically used by Einstein, and it is sometimes the starting point today.
As Einstein himself later acknowledged, the derivation of the Lorentz transformation tacitly makes use of some additional assumptions, including spatial homogeneity, isotropy, and memorylessness.
A more modern example of deriving the Lorentz transformation from electrodynamics (without using the historical aether concept at all), was given by Richard Feynman.
[5] George Francis FitzGerald already made an argument similar to Einstein's in 1889, in response to the Michelson–Morley experiment seeming to show both postulates to be true.
He wrote that a length contraction is "almost the only hypothesis that can reconcile" the apparent contradictions.
Lorentz independently came to similar conclusions, and later wrote "the chief difference being that Einstein simply postulates what we have deduced".
It has often been argued (such as by Vladimir Ignatowski in 1910,[6][7][8] or Philipp Frank and Hermann Rothe in 1911,[9][10] and many others in subsequent years[11]) that a formula equivalent to the Lorentz transformation, up to a non-negative free parameter, follows from just the relativity postulate itself, without first postulating the universal light speed.
The numerical value of the parameter in these transformations can then be determined by experiment, just as the numerical values of the parameter pair c and the Vacuum permittivity are left to be determined by experiment even when using Einstein's original postulates.
A break in Einstein's logic occurs where, after having established "the law of the constancy of the speed of light" for empty space, he invokes the law in situations where space is no longer empty.
This would be equivalent to stating that we know that the introduction of matter into a region, and its relative motion, have no effect on lightbeam geometry.
Such a statement would be problematic, as Einstein rejected the notion that a process such as light-propagation could be immune to other factors (1914: "There can be no doubt that this principle is of far-reaching significance; and yet, I cannot believe in its exact validity.
It seems to me unbelievable that the course of any process (e.g., that of the propagation of light in a vacuum) could be conceived of as independent of all other events in the world.
")[13] Including this "bridge" as an explicit third postulate might also have damaged the theory's credibility, as refractive index and the Fizeau effect would have suggested that the presence and behaviour of matter does seem to influence light-propagation, contra the theory.
[14] A similar suggestion that the reduction of GR geometry to SR's flat spacetime over small regions may be "unphysical" (because flat pointlike regions cannot contain matter capable of acting as physical observers) was acknowledged but rejected by Einstein in 1914 ("The equations of the new theory of relativity reduce to those of the original theory in the special case where the gμν can be considered constant ... the sole objection that can be raised against the theory is that the equations we have set up might, perhaps, be void of any physical content.
For if gravitational fields do play an essential part in the structure of the particles of matter, the transition to the limiting case of constant gμν would, for them, lose its justification, for indeed, with constant gμν there could not be any particles of matter.
A theory along the lines of that proposed by Heinrich Hertz (in 1890)[17] allows for light to be fully dragged by all objects, giving local c-constancy for all physical observers.
The logical possibility of a Hertzian theory shows that Einstein's two standard postulates (without the bridging hypothesis) are not sufficient to allow us to arrive uniquely at the solution of special relativity (although special relativity might be considered the most minimalist solution).
"),[18] but dismissed it on the grounds of a poor agreement with the Fizeau result, leaving special relativity as the only remaining option.
Given that SR was similarly unable to reproduce the Fizeau result without introducing additional auxiliary rules (to address the different behaviour of light in a particulate medium), this was perhaps not a fair comparison.
In the rigorous mathematical formulation of special relativity, we suppose that the universe exists on a four-dimensional spacetime M. Individual points in spacetime are known as events; physical objects in spacetime are described by worldlines (if the object is a point particle) or worldsheets (if the object is larger than a point).
In addition to events and physical objects, there are a class of inertial frames of reference.
(In practice, these conversion laws can be efficiently handled using the mathematics of tensors.)
Mathematically, each physical law can be expressed with respect to the coordinates given by an inertial frame of reference by a mathematical equation (for instance, a differential equation) which relates the various coordinates of the various objects in the spacetime.
The second postulate can be used to imply a stronger version of itself, namely that the spacetime interval is invariant under changes of inertial reference frame.
The postulates of special relativity can be expressed very succinctly using the mathematical language of pseudo-Riemannian manifolds.
The first postulate is an assertion that the laws of physics are invariant when represented in any frame of reference for which g is given by the Minkowski metric.
In this theory, the first postulate remains unchanged, but the second postulate is modified to: The physical theory given by classical mechanics, and Newtonian gravity is consistent with Galilean relativity, but not special relativity.
Conversely, Maxwell's equations are not consistent with Galilean relativity unless one postulates the existence of a physical aether.