# 2.3: Additional Postulates

Skills to Develop

- Learn more postulates for spacetime, inertial frames of reference, equivalence of inertial frames and relativity of time

We make the following additional assumptions:

Spacetime is homogeneous and isotropic. No time or place has special properties that make it distinguishable from other points, nor is one direction in space distinguishable from another.^{1}

Postulate 4 (P4): Inertial frames of reference exist

These are frames in which particles move at constant velocity if not subject to any forces** ^{2}**. We can construct such a frame by using a particular particle, which is not subject to any forces, as a reference point. Inertial motion is modeled by vectors and parallelism.

Postulate 5 (P5): Equivalence of inertial frames

If a frame is in constant-velocity translational motion relative to an inertial frame, then it is also an inertial frame. No experiment can distinguish one preferred inertial frame from all the others.

Postulate 6 (P6): Relativity of time

There exist events \(1\) and \(2\) and frames of reference defined by observers \(o\) and \(o'\) such that \(o \perp r_{12}\) is true but \(o' \perp r_{12}\) is false, where the notation \(o \perp r\) means that observer \(o\) finds \(r\) to be a vector of simultaneity according to some convenient criterion such as Einstein synchronization.

Postulates **P3** and **P5** describe symmetries of spacetime, while **P6** differentiates the spacetime of special relativity from Galilean spacetime; the symmetry described by these three postulates is referred to as Lorentz invariance, and all known physical laws have this symmetry. Postulate **P4** defines what we have meant when we referred to the parallelism of vectors in spacetime (e.g., in figure 1.3.2). Postulates **P1-P6** were all the assumptions that were needed in order to arrive at the picture of spacetime described in chapter 1. This approach, based on symmetries, dates back to 1911** ^{3}**. Surprisingly, it is possible for space or spacetime to obey our flatness postulate

**P2**while nevertheless having a nontrivial topology, such as that of a cylinder or a Möbius strip. Many authors prefer to explicitly rule out such possibilities as part of their definition of special relativity.

### References

- For the experimental evidence on isotropy, see http://www. edu-observatory.org/physics-faq/Relativity/SR/experiments.html\ #Tests_of_isotropy_of_space
- Defining this no-force rule turns out to be tricky when it comes to gravity. As discussed in ch. 5, this apparently minor technicality turns out to have important consequences.
- W. v. Ignatowsky, Phys. Zeits. 11 (1911) 972. English translation at en.wikisource.org/wiki/Translation:Some_General_Remarks_on_ the_Relativity_Principle

### Contributor

- Benjamin Crowell (Fullerton College). Special Relativity is copyrighted with a CC-BY-SA license.