The 14 postulates of Special Relativity

[grav-universe]

These are really postulates of physics in general, but is aimed mostly toward the dynamics of space and time and of motion, specifically to that which is necessary to derive Special Relativity. This is not a theory, but a collection of postulates which serve as the foundation of many such theories, put together here in the fashion that they might normally be used in physics. As such, this thread is of course open to any discussion as to how they might be made more rigorous or to the consideration of any further postulates that might be added.

CONDITIONS

1) Local - It will be assumed that identical experiments can and will be performed under identical conditions at any location in any inertial frame, locally to the experiment, as with setting temperature for example, by applying the same local methods for achieving these conditions.

2) Ideal - It is assumed that the effects of any phenomenon that cannot be locally controlled and which might affect the experiment in some way but are not meant to do so, such as cosmic background radiation or universal expansion, will be negligible.

SPACE

3) Homogeneity - Any identical experiments performed under the same conditions and with the same dynamics applied in any direction relative to an inertial observer at any location in any inertial frame will attain the same results according to that observer.

4) Volume - If a large container of any shape is filled separately with two different sets of naturally occurring physical entities that take up space, such as between different types of atoms, then the average number ratio or proportion between those two sets will remain the same at any location in any inertial frame when placed under the same conditions.

5) Rigidity - Any sturdy materials, regardless of composition, will retain the same proportions to each other no matter which direction they are turned locally within a frame.

Test - This can easily be tested by placing two rigid rods of different composition side by side and cutting them to the same length, defined simply by having their ends meet, then turning them in various directions to test whether the ends no longer both meet.

A) Coordinate choice for rulers - As per postulate 5, even if rods were to contract or expand in different directions, they would all do so in the same way regardless of composition, so we could not directly measure a difference, so we will consider them to be rigid and to remain the same length in any direction. As per postulate 4, we will make a ruler according to a particular number of such physical entities placed one after another along a straight line (a staight line to be defined), and this shall be done the same in every frame.

GEOMETRY

6) Circle - In accordance with the third postulate of Euclidean geometry, a circle can be described with any center and radius.

B) Coordinate choice for spheres - We will apply postulates 5 and 6 to extend to spheres, whereby after identifying two random points upon a rigid body of any shape, we can then rotate the body in all directions about one of the points and mark the places in space that the second point coincides, which will enclose the surface of a sphere.

7) Line - In accordance with the first postulate of Euclidean geometry, a straight line can be drawn between any two points.

C) Coordinate choice for lines - We will define a straight line as the shortest distance between two points. Since distance is determined by a ruler and we have not yet determined how to make a straight line ruler to begin with yet, the procedure is as follows. We will identify two points in space which will be the endpoints of our ruler. We will then make many identical infinitesimal spheres which have been fashioned in the manner described by coordinate choice B and connect them along their outer surfaces as would be a string of beads. The path that lies between the two points in space that can be found by placing the least number of spheres between them will define a straight line between those two points, and that shall also be the edge of our ruler.

TIME

8) Periodicity - All physical processes, whether they be periodic as with atomic vibrations or a measure of change as with radioactive decay, for examples of such, will occur at the same rate locally in proportion to each other at any location in any inertial frame.

9) Locality - Identical physical processes will occur at the same rate at any location within the same inertial frame.

Test - We can test this by having two sets of identical periodic processes, then moving one to another location within the same frame while continuing to count its periodic rate of each, and after some time has passed, moving the other set to the same location in the same manner as the first while still continuing to count the periodic rates while doing so, and determining that the same count has occurred for both sets.

D) Coordinate choice for clocks - In accordance with postulates 8 and 9, since all local periodic processes occur at the same rate in proportion to each other, and since clocks should connect with these processes in order to have any physical meaning, we will set the timing of clocks in accordance with local natural periodic processes.

MOTION

10) Dynamics - An inertial body will travel at a constant and steady pace unless otherwise acted upon, as observed by an inertial observer.

11) Path - An inertial body will continue to travel in the same direction along a straight line as observed by an inertial observer.

LIGHT

12) Aberration - If a light pulse is emitted from each of two sources as they coincide at the same point in space and the light pulses then travel to an observer at any other location in any frame, that observer will receive both pulses simultaneously, regardless of the motions of the sources.

13) Isotropy - It is assumed that the speed of light is not directly affected in any measurable way by any discernable medium. As such, postulates 2 and 3 will apply, whereby if a clock is placed at the end of a rigid rod and a light pulse is propagated from the clock to the other end of the rod and back, it will do so in the same two way time according to the difference in readings upon the clock, regardless of the direction of the rod in space.

14) Speed - When the lengths of identical rigid rulers and the timing of clocks are set according to the natural physical processes as per coordinate choices A and D, then by applying the method as described in postulate 13, the same difference in readings will be found to pass upon the clock at the end of the rod at any location in any inertial frame.

 

[Naty1]

Seems like a rather long list. What is your purpose?


Why not use just two:

http://en.wikipedia.org/wiki/Postulates_of_special_relativity

...relativity of inertial frames and the fixed speed of light.

 

[Naty1]

I happened to stop here and think about:

Rigidity - Any sturdy materials, regardless of composition, will retain the same proportions to each other no matter which direction they are turned locally within a frame.

(proportions of what??)

Test - This can easily be tested by placing two rigid rods of different composition side by side and cutting them to the same length, defined simply by having their ends meet, then turning them in various directions to test whether the ends no longer both meet.

It doesn't seem accurate (aside from imprecise phrasing) ...but is confusing...rods WILL change length in distant inertial frames depending on orientation...Lorentz contraction...

 

[grav-universe]

Seems like a rather long list. What is your purpose?


Why not use just two:

http://en.wikipedia.org/wiki/Postulates_of_special_relativity

...relativity of inertial frames and the fixed speed of light.

I wanted to include every aspect which is postulated in physics that leads up to the derivation of SR. The second postulate of SR, for example, is actually a composition of three separate postulates, although all related to light.

 

[grav-universe]

I happened to stop here and think about:


It doesn't seem accurate (aside from imprecise phrasing) ...but is confusing...rods WILL change length in distant inertial frames depending on orientation...Lorentz contraction...

Yes, I thought about that also after I posted. By "locally within a frame", I mean stationary to an observer.

 

[Phrak]

Seems like a rather long list. What is your purpose?

Why not use just two:
http://en.wikipedia.org/wiki/Postulates_of_special_relativity
...relativity of inertial frames and the fixed speed of light.

The list is substantially we include the postulates hiding in plain sight:

3. Space is isotropic.
4. Space is homogeneous.

The postulates of arithmetic
We'd have to add more to deal with algebra.
And more to associate measurements with numbers.
And the dozens of others I can't think of at the moment.

 

[bcrowell]

The postulates of arithmetic
We'd have to add more to deal with algebra.

Actually the postulates of Euclidean geometry are sufficient to build up the entire real number system. This is how Euclid originally treated the real number system. If you read the Elements, a lot of it is actually about what we would today refer to as algebra and number theory. Since the OP is essentially including Euclid's axioms in the list, it's not necessary to add separate axioms for the number system. (From the modern point of view Euclid's axioms are incomplete, e.g., they need to include something specifying the completeness property of the reals.)

On the other hand, I'm not sure why the list includes some of Euclid's axioms and not others. I think the list, as written, is too weak to rule out a space of uniform, nonzero curvature.

The use of infinitesimals in 7C is interesting. Infinitesimals can be treated rigorously in an axiomatic system. The distinction between the standard reals and a system that includes both the reals and infinitesimals (Abraham Robinson's hyperreals) can only be expressed in second-order logic; in first-order logic, both are equivalent to Euclidean geometry on a line. The idea you're trying to express, of partitioning a line into infinitesimal points, can be formalized. See ch. 4 of this book http://www.math.wisc.edu/~keisler/calc.html for a nontechnical development of it. Since the flavor of your axioms is physical, I don't think it's a big deal that you're not explicitly developing all this.

The notion of an inertial observer hasn't been defined in 10 and 11.

12 seems too strong to me. It rules out SR in a space with zero curvature and a cylindrical topology.

I'd consider 2 unnecessary. We don't expect that such an axiomatic system will apply without approximation to the real universe. The purpose of such an axiomatization is to spell out a logical system that is consistent within itself.

As a matter of taste, I dislike the fact that your system singles out light for a special role. For systems that don't assign a special role to light, see:
Morin, Introduction to Classical Mechanics, Cambridge, 1st ed., 2008
Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51
Morin and Rindler aren't very explicit about the exact axiomatization they're using. For a more explicit axiomatization:
http://www.lightandmatter.com/html_books/6mr/ch01/ch01.html

 

[bcrowell]

A few more thoughts: --

You're trying to build up a lot of the geometric apparatus by hacking around with the Euclidean axiomatization. You're reinventing the wheel. The ideas about coordinates that you're trying to express can be handled using a standard lattice construction in affine geometry: http://www.lightandmatter.com/html_books/genrel/ch02/ch02.html#Section2.1 The classic textbook you really want to read is Coxeter's Introduction to Geometry. (It's not an "introduction" at the usual high school level. It's an introduction more at the level of upper-division math majors.)

You're trying to describe flat spacetime. 7 and 12 rule out some curved spacetimes but not others, and allow some flat spacetimes but not others. You could include something like the parallel postulate. But since you want to introduce coordinatization anyway, you could just incorporate the axioms of affine geometry into your list, and then I think you'd be all set. Affine geometry is a flat-space geometry.

It's not clear to me where in the list you set the dimensionality of spacetime to be 3+1. In a really elegant axiomatization, one would want to make all the axioms independent of the dimensionality, except for one that sets it to 3+1.

 

[Fredrik]

Bcrowell:

He probably shouldn't assume that it's 3+1, because the +1 implies a Lorentzian metric, and that's one of the things that these "derivations" are trying to prove. But it's certainly necessary to assume that the underlying set of the mathematical structure that we're going to use as a model of space and time is $$\mathbb R^4$$.

Everyone:

Special relativity is a theory of physics that tells us how to interpret the mathematics of Minkowski spacetime (and mathematically defined particles and fields in it) as predictions about results of experiments. It doesn't need to be "derived". If you for some reason still want to do it, there are a few things you need to keep in mind.

What does it even mean to "derive" SR? These are the only possibilities I can see:

• To assume that spacetime is $$\mathbb R^4$$, with the standard vector space structure and equipped with a bilinear form, and prove that the bilinear form must be the Minkowski form.
• To assume that spacetime is $$\mathbb R^4$$, with the standard manifold structure and equipped with a metric, and prove that the metric must be the Minkowski metric.
• To assume that the set of bijections on $$\mathbb R^4$$ that have the properties that we expect functions that change coordinates from one inertial frame to another to have, is a group, with composition of functions as the "multiplication", and prove that this group must be the Poincaré group.

To do any of these proofs, you have to start with some set of mathematical axioms. (The assumptions I already mentioned in my list are clearly not enough). The "postulates" that people usually start with aren't even mathematical statements, so they can't be used as the starting point of a proof. It's never possible to derive a mathematical theorem from non-mathematical postulates. You have to find mathematical statements that correspond to the loosely stated ideas in your list of "postulates", and use them as the axioms.

It's much more important to get your list of mathematical axioms exactly right than to get the "postulates" right.

 

[bcrowell]

To do any of these proofs, you have to start with some set of mathematical axioms. (The assumptions I already mentioned in my list are clearly not enough). The "postulates" that people usually start with aren't even mathematical statements, so they can't be used as the starting point of a proof. It's never possible to derive a mathematical theorem from non-mathematical postulates. You have to find mathematical statements that correspond to the loosely stated ideas in your list of "postulates", and use them as the axioms.

You're entitled to your opinions about the relationship between mathematics and physics, but I think you're way off base here. Physicists simply don't mean the same thing mathematicians do when they talk about axioms and proofs. By your assertion, the following work by Einstein would appear to be worthless:
- "On the electrodynamics of moving bodies," http://www.fourmilab.ch/etexts/einstein/specrel/www/ , section I.1, axiomatizes the behavior of clocks
- In the same paper, section I.2 axiomatizes the behavior of spacetime

It's much more important to get your list of mathematical axioms exactly right than to get the "postulates" right.

Absolutely wrong. It's important to get your physical assumptions right. The mathematical formalism is secondary.

 

[Fredrik]

By your assertion, the following work by Einstein would appear to be worthless:
- "On the electrodynamics of moving bodies," http://www.fourmilab.ch/etexts/einstein/specrel/www/ , section I.1, axiomatizes the behavior of clocks
- In the same paper, section I.2 axiomatizes the behavior of spacetime

The first paper ever written on special relativity certainly wasn't worthless, but that doesn't mean we can't find a better way to express those ideas a hundred years later.

Absolutely wrong. It's important to get your physical assumptions right. The mathematical formalism is secondary.

Do you realize that we're talking about "postulates" that are only used to find a mathematical structure that might be able to replace Galilean spacetime as a model of space and time? The postulates aren't part of the definition of the theory, so they aren't important in any way, except historically, and if it's history we're interested in, there's no reason to try to improve Einstein's original list of postulates.

The only reason I can think of to add more postulates would be if we're trying to find rigorously provable relationships between mathematical statements. I don't see the point of trying to improve Einstein's original work if we're going to do a sloppy "physicist's derivation" anyway.

SR is defined by a completely different set of axioms, which tells us how to interpret the mathematics of Minkowski spacetime as predictions about results of experiments. Some details in the mathematics are unimportant (e.g. do we want to define spacetime with a manifold structure or a vector space structure), but it's certainly fair to say that the actual theory is more important than the loosely stated ideas that were used to find it.

 

[grav-universe]

Thanks for your comments, everyone. :)

bcrowell, it looks like we are on the same page here. I thought about including all five Euclidean postulates, but didn't want to be too redundant. I figured that the fourth Euclidean postulate, that all right angles are equal to each other, would basically be saying that straight lines will remain the same whatever direction they are turned within a frame, which follows for rods and physical objects, which was already covered in the postulates about space in the OP. Likewise, I realized that the second postulate, that a line can be extended indefinitely, was basically covered with the 11th postulate of the OP, whereas an object traveling inertially will continue to travel a straight line, so these Euclidean postulates have already manifested themselves physically.

I wasn't sure how the parallel postulate would directly apply, however, so didn't include it, although it might be useful to have a definition describing the construction of parallel lines themselves. I'm also not sure what you mean about the 12th postulate of the OP. If two sources each emit a light pulse when the sources coincide in the same place, then regardless of the motions of the sources or the topology, the two light pulses would travel together while following the same path and reach a destination simultaneously.

I agree with what your saying about the mathematics. It would just be a matter of applying a coordinate system according to what the postulates have already provided. That is also in line with what you were saying about the 3+1, though, so I suppose one would also somehow have to postulate that three coordinates are necessary to identify a point in space, including something along the lines of having axes at three right angles to each other, although I have not done this, so it is something to work on. :)

Once postulated, we could define the axes of our system by turning three identical straight line rods until the end of each rod is equidistant from the ends of the other two rods, the intersection of which is the origin. We then set the clocks according to the physics of each frame, and since the timing is the same anywhere within a frame, that is 3+1 for the frame. From there, it is simply a matter of synchronizing the clocks using light, and from the last three postulates, we know that any frame will find that 2AB / (tA' - tA) = c for the two way speed of light in any direction, so that it is also possible to synchronize two clocks to AB / (tB - tA) = AB / (tA' - tB) = c in any direction. And of course the derivation of SR follows from there.

 

[grav-universe]

The notion of an inertial observer hasn't been defined in 10 and 11.

Oh yes, I should have defined that better. An inertial body as used here is inactive and is not affected by external influences. The same applies for an inertial observer.