Mermin's geometrical approach to SR

[bcrowell]

N. David Mermin has an interesting geometrical approach to SR that I came across today. He seems to have described it in the following places:

1. Mermin, N. David, "Space-time intervals as light rectangles," Amer. J. Phys. 66 (1998), no. 12, 1077
2. a popular-level book called "It's About Time"
3. various pdf's of talks, linked to from http://people.ccmr.cornell.edu/~mermin/homepage/ndm.html (minkowski.pdf being the most detailed)

One thing I hadn't realized before is that there is a connection between the spacetime interval and the area of rectangles with lightlike edges. There are straightforward arguments that area in the x-t plane has to be conserved under boosts in the x direction, regardless of whether you're talking about SR or Galilean relativity or any other theory that doesn't violate the homogeneity and isotropy of spacetime (http://www.lightandmatter.com/area1book6.html , appendix 1) . Since area is conserved, and the spacetime interval can be interpreted as an area in the case of SR, you get a very simple and straightforward geometrical argument that the interval is invariant.

He also has a cute way of getting at the relativistic combination of velocities via the Doppler shift, which I think is far more transparent than any other approach I've seen. Essentially you can get the velocity w that results from combining velocities u and v by setting the cumulative Doppler shift equal to the product of the two partial Doppler shifts:
$$\sqrt{\frac{1-w}{1+w}} = \sqrt{\frac{1-u}{1+u}} \sqrt{\frac{1-v}{1+v}}$$
This can be solved for w to get the usual formula, or if you take logs you get the additive relation between rapidities. But he has a cute way of looking at this purely geometrically (p. 25 of minkowski.pdf), which can then be used to derive the Lorentz gamma factor from Einstein's 1905 axiomatization of SR.

Personally I don't think the 1905 axiomatization is the right one to use in the year 2010, but a lot of Mermin's stuff nevertheless fits very nicely with my own preferred way of teaching SR, which is also heavily geometrical ( http://www.lightandmatter.com/html_books/6mr/ch01/ch01.html ).

 

[robphy]

Here are some related works (including Mermin's) that recognize the invariance of the area:

https://www.physicsforums.com/showthread.php?t=205509&page=3
which links to
https://www.physicsforums.com/showthread.php?p=747083#post747083

The velocity composition from the multiplication of Doppler Effect factors
is a featured calculation using the Bondi k-calculus.

My user-avatar features an invariant volume (the intersection of the future and past light cones of two timelike related events).

 

[bcrowell]

Thanks for the links, robphy!

I'm surprised that all of these papers treat frame-invariance of x-t areas under x-boosts as a relativistic fact, when actually it's not specifically relativistic.

 

[robphy]

Thanks for the links, robphy!

I'm surprised that all of these papers treat frame-invariance of x-t areas under x-boosts as a relativistic fact, when actually it's not specifically relativistic.

Of course, these papers are not just about the areas... or just some arbitrary area.
It's really about the area of a triangle whose legs are lightlike (the size of the legs are essentially the light-cone coordinates). So, its about the invariance of both the area and the lightlike directions.

Consider a timelike vector decomposed into the sum of two future-lightlike vectors.
This is such a triangle. Under a boost, the tip of that timelike vector traces out a hyperbola (since of course its Minkowski norm is invariant under the boost), which implies that the area of that triangle is invariant.

Have a look at page 8 of my paper http://arxiv.org/abs/physics/0505134 (url corrected)

 

[bobc2]

RobPhy, excellent paper. The strong geometric emphasis seems to suggest a physical reality of a 4-dimensional universe. Do you think Einstein had this in mind when he wrote the comments to the wife of his close friend Besso just after Besso's death? I imagine that Einstein had conversations with Kurt Godel on this subject on some of the many walks home together from the Princeton campus at the end of the day. I think Godel had strong opions about the reality of four spatial dimensions (the "Block Universe"?). Of course Max Tegmark sees the universe as strictly a mathematical construct.

Some physicists today seem to adapt a view of a real 4-D universe, whereas others seem to back away from that conclusion, using the imaginary time (X4 = ict) as their red flag.

How do you (you also, bcrowell) regard the question of real 4 spatial dimensions?

It might be interesting for someone to set up a poll of forum SR participants: Vote yes or no, whether you accept a literal 4-dimensional (spatial) universe.

 

[bcrowell]

Of course, these papers are not just about the areas... or just some arbitrary area.
It's really about the area of a triangle whose legs are lightlike (the size of the legs are essentially the light-cone coordinates). So, its about the invariance of both the area and the lightlike directions.

I understand that. What I'm pointing out that you may not have realized is that the area in the x-t plane is conserved under boosts simply based on the symmetry properties of spacetime, regardless of whether you're talking about SR or Galilean relativity. I consider this important because it provides a way to derive the Lorentz transformation from the invariance of area.

 

[grav-universe]

He also has a cute way of getting at the relativistic combination of velocities via the Doppler shift, which I think is far more transparent than any other approach I've seen. Essentially you can get the velocity w that results from combining velocities u and v by setting the cumulative Doppler shift equal to the product of the two partial Doppler shifts:
$$\sqrt{\frac{1-w}{1+w}} = \sqrt{\frac{1-u}{1+u}} \sqrt{\frac{1-v}{1+v}}$$

As an aside, that basic relation between the Doppler shifts is fundamentally true regardless of the kinematic theory involved, with arbitrary coordinatization of the frames, regardless of how the frames are synchronized, or even what the particles are or their speed that are used to find the Doppler shifts.

Let's say that from reference frame A, arbitrarily coordinated and synchronized, observers in frame A measure another set of observers to be traveling at speed v in one direction along the x-axis in frame B and a third set of observers to be traveling in the opposite direction along the x-axis at speed u in frame C. Frames B and C are also arbitrarily coordinated and synchronized. Frame B emits electrons at regular intervals of time tB according to frame B. Frame A receives them at regular intervals of tA according to A's own clocks, so the Doppler shift from B to A is Dv = (frequency observed) / (frequency emitted) = (1 / tA) / (1 / tB) = tB / tA. Then frame A re-emits them to frame C at the same speed and at the same intervals at which they were received, and frame C receives them at regular intervals of tC according to C's clocks, so the Doppler shift from A to C is Du = tA / tC. If B were to send the electrons directly to C, then we would have Dw = tB / tC, and we can see that frame A receiving and then immediately re-emitting the particles in the same way is the same as B sending the particles directly to C also, whereas Dw = Du Dv.

 

[robphy]

I understand that. What I'm pointing out that you may not have realized is that the area in the x-t plane is conserved under boosts simply based on the symmetry properties of spacetime, regardless of whether you're talking about SR or Galilean relativity. I consider this important because it provides a way to derive the Lorentz transformation from the invariance of area.

Yes, I agree. Minkowski, Galilean, and Euclidean geometries are affine geometries.
The area is preserved by boost/"rotation"-transformations, which have unit determinant, independent of the signature.

In SR, the legs of the triangle (or sides of the "light-rectangles") are proportional to the eigenvalues of the boost (the Doppler factor k and (since det R=1) 1/k [or exp(rapidity) and exp(-rapidity)]) with the eigenvectors along the invariant lightlike directions. (In the Galilean case, there's only one eigenvalue, 1, for the invariant spacelike direction... and none for the Euclidean space, leaving only the origin invariant.)

With a little algebra,
gamma=cosh(rapidity)=( k+(1/k) )/2
beta*gamma=sinh(rapidity)=(k-(1/k))/2)
and beta=tanh(rapidity)=( k+(1/k) )/( k-(1/k) )

These results come from radar methods and Bondi's k-calculus (which is sadly absent from many relativity books).

https://www.physicsforums.com/showthread.php?p=934989#post934989 (radar method)
https://www.physicsforums.com/showthread.php?t=117439 (k-calculus)

 

[robphy]

Oops
beta=tanh(rapidity)=( k - (1/k) )/( k + (1/k) )

 

[jason12345]

It's all very interesting David mermin's and others, approach to SR, but are they looking at these other approaches simply because they're different?

I've never come across any approach that has genuinley simplified Einsten's axioms:

1. There is a max velocity c for all moving obects

2. The laws of physics are the same in all frames.

Usually, they replace 1 with some other postulate that itself is derivable from 1 and 2.

 

[bcrowell]

It's all very interesting David mermin's and others, approach to SR, but are they looking at these other approaches simply because they're different?

I've never come across any approach that has genuinley simplified Einsten's axioms:

1. There is a max velocity c for all moving obects

2. The laws of physics are the same in all frames.

Usually, they replace 1 with some other postulate that itself is derivable from 1 and 2.

Hmm...I notice that your #1 is different from Einstein's #1. Einstein's #1 refers specifically to electromagnetic waves.

Personally the reasons I prefer the approach I use to approaches based on the 1905 Einstein axiomatization are:

1. With the benefit of a century's hindsight, we know that the electromagnetic field is not the only fundamental field, and therefore it shouldn't play any special role in relativity. We see the negative effect of this misplaced special role all the time on this forum. E.g., people want to know why c appears in E=mc^2. That's a heck of a good question for a beginner who has only seen c enter into SR as the "speed of light."
2. In 1905 Einstein didn't know that GR would be a geometrical theory. Now that we know that, it makes sense to look at SR geometrically as well.
3. The modern point of view is that relativity is basically a theory of the possible causal relations between events. Our approach to teaching the subject should reflect that.

You mention simplicity as a criterion for picking an axiomatization. One thing to watch out for is that Einstein's 1905 axiomatization may appear simpler than later axiomatizations simply because he left out axioms that are in fact logically necessary, but that he didn't care to focus on. For example, in the 1905 paper he specifically appeals to homogeneity of spacetime: "In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time." Since he didn't have GR yet, he didn't know that this was actually quite a strong assumption: flat spacetime. Elsewhere he says, "We assume that [...] If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other." This is essentially a restriction to nonrotating frames (but again, he didn't know that in 1905). I'm also pretty sure that his treatment implicitly assumes isotropy of spacetime, although he doesn't specifically say so. So at this point we have at least five axioms in the 1905 paper:

1. Clock synchronization is transitive.
2. Spacetime is homogeneous.
3. Spacetime is isotropic.
4. Uniform motion is relative.
5. The speed of electromagnetic waves is frame-independent.

This isn't any shorter or simpler than other axiomatizations.

-Ben

 

[Cleonis]

In 1905 Einstein didn't know that GR would be a geometrical theory. Now that we know that, it makes sense to look at SR geometrically as well.

I very much agree with you.
One way of expressing that notion: the transition from Newtonian to SR and the transition from SR to GR have the following in common: the metric of spacetime was replaced.

SR is usually introduced by asserting invariance of the speed of light followed by a formal derivation of logical implications. As you point out: this has the disadvantage that a beginner is badly wrongfooted into thinking that the whole thing is down to properties of light.

I much prefer an approach in which the invariance of the speed of light is a theorem of SR, rather than an axiom.

A metric based exposition of SR first introduces the concept of a metric for Newtonian space-and-time.
Then it is asserted that the homogeneity and isotropy of inertia is one and the same thing as the homogeneity and isotropy of space-and-time.
In other words, you assert that Newton's First law is to be reinterpreted as a statement about space-and-time. The nature of space-and-time is such that an object in inertial motion will cover equal distances in equal intervals of time.

(Newton's first law is commonly presented as an assertion about objects, rather than as an assertion about the physical properties of space-and-time. This has the severe disadvantage of wrongfooting the student.)For SR the first law doesn't need to be rephrased: you assert: an object in inertial motion covers equal distances in equal intervals of time.
Given that in SR you work with coordinate distance and coordinate time the amounts of coordinate distance and coordinate time are frame dependent, but regardless of that the property 'equal distances are covered in equal intervals of time' applies for Minkowski spacetime. This is embodied by the metric.

As you point out: a lot of the invariance concepts can be introduced independently of SR.
I think beginners are under the impression that they are supposed to disregard space, since in SR there is no such thing as assigning an absolute velocity vector to an object, a velocity wrt a background reference such as a Lorentz ether.

The result, I think, is that instead of trying to think in terms of spacetime beginners steadfastly avoid thinking in terms of spacetime.

 

[Cleonis]

One thing to watch out for is that Einstein's 1905 axiomatization may appear simpler than later axiomatizations simply because he left out axioms that are in fact logically necessary, but that he didn't care to focus on.

I agree.

Elaborating:
In physics there is something of a tradition of presenting a new theory in the form of axioms or postulates, but it's really quite different from presenting axioms in mathematics.

In mathematics the requirement is that the set of axioms is exhaustive. In mathematics that is a realistic demand. In physics, if you would try to present an exhaustive set your presentation would be cluttered.

In physics, the underlying purpose of the "axiomatic" presentation is to be evocative. The statements that are presented are the ones that the author regards as the core issues.As I argued in my previous posting, Newtonian theory of motion and SR have the following principle in common: 'objects in inertial motion cover equal distances in equal intervals of time.'
(I like the similarity with Kepler's law of area's: equal areas are swept out in equal intervals of time.)

It should be possible to formulate two sets of axioms, one for Newtonian and one for SR, that differ in only one axiom. That serves to narrow down the choice of axioms.

(Compare the many mathematically equivalent ways to formulate Euclid's fifth axiom. Among those Playfair's axiom is regarded as the most elegant and economical, because with minimal change the axiom is reformulated for the other two geometries.)For Newtonian and SR common axioms are:
- The principle of relativity of inertial motion.
- proper acceleration: F=ma

For Newtonian you get:
space = Euclidean
time = uniform

And for SR you get:
spacetime = Minkowskian