Identifications between the model and the real world

[Fredrik]

I've been saying for years that special relativity can be defined as "the claim that space and time can be represented mathematically by Minkowski space", but this isn't really true. Minkowski space is just a mathematical structure, and as such it doesn't make any predictions about the real world that we can test in experiments. In order to turn the model into a theory of physics, we have to make some identifications between things in the model and things in the real world. For example:

A time-like geodesic = the motion of a massive particle unaffected by forces
A time-like curve = the motion of a massive particle
A null geodesic= the motion of a massless particle
The integral of $\sqrt{-g_{\mu\nu}dx^\mu dx^\nu}$ along a time-like curve = what a clock measures when its motion is described by that curve

Do you know if someone has already made a complete list of the identifications that are necessary? I think it would be interesting to see one. One thing I'm interested in is if there's a set of identifications that defines a theory of physics that's only capable of describing inertial motion. (Does it make any sense at all to say that SR can't handle acceleration even though Minkowski space clearly can?)

If there is no such complete list, maybe we can make one right here. Post what you think should be on it.

Do we need to mention meter sticks, accelerometers and other kinds of measuring devices, or can we (in principle) measure everything with clocks, light, mirrors and photon detectors?

 

[Mentz114]

Fredrik,

I've been saying for years that special relativity can be defined as "the claim that space and time can be represented mathematically by Minkowski space", but this isn't really true.

Why not ? 'Represent mathematically' is exactly what SR does for flat space-time.

Minkowski space is just a mathematical structure, and as such it doesn't make any predictions about the real world that we can test in experiments.

I believe it does. See the sticky thread above.

A time-like geodesic = the motion of a massive particle unaffected by forces
A time-like curve = the motion of a massive particle
A null geodesic= the motion of a massless particle
The integral of along a time-like curve = what a clock measures when its motion is described by that curve

These are all testable !

M

 

[Hurkyl]

I've been saying for years that special relativity can be defined as "the claim that space and time can be represented mathematically by Minkowski space", but this isn't really true. Minkowski space is just a mathematical structure, and as such it doesn't make any predictions about the real world that we can test in experiments.

But the claim that Minkowski space represents space and time does make assertions about the real world. For example, that all physical laws are Poincaré symmetric.

Of course, this really isn't useful until you start making postulates about the other fundamental objects of the physical theory, such as particles and fields.

 

[DrGreg]

Mentz, I think you are missing Fredrik's point. He's not asking for experimental verification. He's asking how to make the link between the mathematical theory of Minkowski space with the physical universe, and he's given some examples in his second paragraph. He's asking for his list to be expanded to a complete list. (I can't do that -- I might be able to add more examples but I couldn't guarantee completeness.)

You don't need metre sticks -- you define distance via radar (for an inertial observer (only)).

You don't need accelerometers -- you define proper acceleration via the co-moving inertial frame and first and second derivatives of distance with respect to time.

However, it's an implicit physical assumption that radar distance is the same as ruler distance (this is essentially the 2nd postulate in disguise) and that proper acceleration is what accelerometers measure (this is related to the clock hypothesis that an accelerating clock ticks at the same rate as a co-moving inertial clock according to the inertial clock).

Having said that, there is the problem of deciding what is an inertial observer in the first place. Mathematically, we can just assume a class of observers who exist and who move at constant velocity relative to each other. In practice, we need to identify such observers, and this means those whose accelerometers read zero! (Which, if you think about it, amounts to saying those for whom Newton's First Law holds.)

 

[Fredrik]

Why not ? 'Represent mathematically' is exactly what SR does for flat space-time.

I'm not saying that the claim "Minkowski space represents space and time mathematically in SR" is wrong. I'm just saying that we need to add more postulates to get a testable theory. For example, by using the Pythagorean theorem on a triangle in Minkowski space, we get a result that we interpret as a prediction about time dilation, but you can't test that prediction until you have decided to trust that a clock in the real world measures something that corresponds to the length of a certain line segment in Minkowski space. That decision adds something new to the theory. It's a postulate about a correspondence between something in the model and something in the real world.

So SR is definitely more than just the mathematics of Minkowski space. In order to use Minkowski space in a theory of physics we have to make certain identifications between the model and the real world, and those identifications are the true postulates of the theory. (The fact that the model is Minkowski space is of course also a postulate of the theory).

Of course, this really isn't useful until you start making postulates about the other fundamental objects of the physical theory, such as particles and fields.

We can introduce matter and interactions using the principle of least action, but we can also do it "manually". We can e.g. postulate that the motion of light is represented by null geodesics and that a reflection against a mirror is represented by a point where one null geodesic ends and another starts. We can postulate that an accelerating rocket is represented by a set of time-like curves that satisfy the requirement of Born rigidity. Etc.

Having said that, there is the problem of deciding what is an inertial observer in the first place. Mathematically, we can just assume a class of observers who exist and who move at constant velocity relative to each other. In practice, we need to identify such observers, and this means those whose accelerometers read zero! (Which, if you think about it, amounts to saying those for whom Newton's First Law holds.)

This is how I would define an inertial frame mathematically: An inertial frame is a global coordinate system x, such that

(i) The preimage of the 0 axis under x is a time-like geodesic.
(ii) The preimages of the 1-3 axes under x are space-like geodeics.
(iii) If p is an event on a null line through $x^{-1}(0)$, then $x(p)^T\eta x(p)=\eta$.
(iv) $g(\partial_\mu,\partial_\nu)=\eta_{\mu\nu}$

(Remove the word "global" and the definition holds in GR too, I think).

I agree that "observers whose accelerometers read zero" is the appropriate definition of an inertial observer in the real world. I'm not sure how to define an accelerometer in the real world though. I think we can do that with light too, e.g. by attaching a bunch of (identical) clocks, lasers and photon detectors to a rigid frame and have the clocks use the lasers to compare their relative ticking rates.

 

[Mentz114]

Fredrik,

So SR is definitely more than just the mathematics of Minkowski space. In order to use Minkowski space in a theory of physics we have to make certain identifications between the model and the real world, and those identifications are the true postulates of the theory. (The fact that the model is Minkowski space is of course also a postulate of the theory).

OK, I'm thinking about it, but I'm having trouble because I think Minkowski spacetime, described in words is SR.

Anyhow, I apologise for the tone of my earlier post, clearly I have misunderstood.

M