Relativity using the Bondi k-Calculus - Comments

[robphy]

robphy submitted a new PF Insights post

Relativity using the Bondi k-Calculus

Continue reading the Original PF Insights Post.

 

[houlahound]

Great insight.

 

[houlahound]

To the author, you have exposed different approaches to this body of knowledge. What approach or ways would you use and sequence say for undergraduate instruction.

 

[robphy]

Any approach I use must use the spacetime diagram
because I think it is difficult to represent the relativity-of-simultaneity using boxcars as "moving frames of reference".

Any approach I use must use radar methods
to motivate measurements and the assignment of coordinates.
I think radar methods are more straightforward than lattices of "clocks" and "rods".
(For inertial motions in special relativity, they are equivalent.
However, for more general motions in special and general relativity, they may differ...
and would require more advanced discussion to address.)

In my opinion, the Bondi k-calculus method (with its emphasis on radar measurements) is the best starting point, especially for algebra-based physics. With the k-calculus methods, the standard textbook formulas are straightforward to derive and fall out naturally.

A related but even less well known approach by Geroch (in his General Relativity from A to B) is also a good starting point. Geroch uses radar methods to emphasize the square-interval and give operational interpretations of the geometry of spacetime (e.g., what simultaneity means to an observer) in both Special Relativity and Galilean Relativity. My AJP article (which inspired the Insight https://www.physicsforums.com/insights/relativity-rotated-graph-paper/ ) was my attempt to combine Bondi's and Geroch's approaches.

From here, I would go on to develop the geometry of Minkowski spacetime, while comparing and contrasting with Euclidean geometry, using the [unappreciated] geometry of Galilean spacetime (e.g., https://www.desmos.com/calculator/ti58l2sair ... play with the E-slider) ...something I call "Spacetime Trigonometry", a large ongoing project with many aspects which generates lots of posters for me at AAPT meetings. (I should really write this up soon... but it would have to be broken into a series of AJP articles.) These are examples of Cayley-Klein geometries, which includes the deSitter spacetimes.. This "unification" can help formalize the numerous analogies mentioned in the literature. In addition, I can develop vector and tensorial methods (algebraically, graphically, and geometrically) in order to make contact with traditional intermediate and advanced presentations of relativity.

 

[houlahound]

Please keep us posted, I will enjoy following your work.

 

[pervect]

Thanks for posting this. A lot of times I want to refer people to Bondi's approach, as I also feel it's one of the best elementary treatments for the person new to relativity. I can and do refer interested people to his book, but it's nice to have a more accessible source.

 

[robphy]

Thanks. I was torn between making it as elementary as possible for a beginner (which would only be a tweak on Bondi or just the equations already provided by Wikipedia) or making clarifications and connections to geometry (the logical next step).

 

[skanskan]

Why does he assume the velocity of light is the same for all inertial observers?

 

[Ibix]

Why does he assume the velocity of light is the same for all inertial observers?

About forty years before Einstein, Maxwell published equations describing electromagnetism. One solution to the equations was a wave, which turned out to have the properties of light. One weird thing was that the speed of the wave came out the same always. Naturally everyone assumed that the equations weren't quite right and the hunt was on to find the problem.

The next forty years were a bit confusing as no one could find anthing wrong. Experiments that were expected to help (e.g. Michelson and Morley) didn't work as predicted, but did provide some ad hoc patches. Einstein had the insight that if the (apparently daft) prediction that light always travels at the same speed for all inertial observers was correct then he could explain all of the confusion. So he made the assumption.

 

[bahamagreen]

Alice's movie is seen by Bob to be in slow motion, and Bob's movie is seen by Alice to be in slow motion. That is similar to SR in which Alice's and Bob's clocks would measure to each other to run slow ... but all your diagrams are presenting the case of increasing separation of the inertial travelers.

To the degree that the diagram tends to suggest that movie duration is a proxy for time dilation... it looks like it only works with cases of increasing separation, not cases of approach. Students would notice this...

If these movies were youtube videos, there would be a time indicator rolling at the bottom of the screen, so for example, both Alice and Bob could see that Alice's movie indicates that it starts at 00:00:00 and increments to 00:60:00 at the end. Although Bob can't necessarily "view Alice's clock" , he can see by the video time index that in comparison to his own clock her video is running slow... suggesting that her time is slower relative to his (and likewise his to hers when he sends video to her).

When Alice and Bob approach each other, it looks like Bob is going to see Alice's movie running faster (shorter time), and Bob's movie will be seen by Alice to be running faster... so this is not similar to SR which would maintain that each measure each others clocks running slow.

 

[Mister T]

To the degree that the diagram tends to suggest that movie duration is a proxy for time dilation... it looks like it only works with cases of increasing separation, not cases of approach. Students would notice this...

That's difference between "see" and "observe". We see Doppler shifted light as it enters our eyes in the same way as we see the movie running slow as its images enter our eyes.

But if you want to observe what is really happening you have to allow for the light travel time. That will lead you to time dilation.

Note that even for the case of increasing separation the time dilation factor is not the same as the Doppler factor.

If the relative speed is $\beta$ then the time dilation factor is $(1-\beta^2)^{\frac{1}{2}}$ whereas the Doppler factor is $\big(\frac{1+\beta}{1-\beta}\big)^{\pm\frac{1}{2}}$.

 

[robphy]

Alice's movie is seen by Bob to be in slow motion, and Bob's movie is seen by Alice to be in slow motion. That is similar to SR in which Alice's and Bob's clocks would measure to each other to run slow ... but all your diagrams are presenting the case of increasing separation of the inertial travelers.

To the degree that the diagram tends to suggest that movie duration is a proxy for time dilation... it looks like it only works with cases of increasing separation, not cases of approach. Students would notice this...

These viewings of movies are not proxies for time-dilation... they are descriptions of the Doppler effect for light.
For observers receding from each other, each observes a "redshift" (or, in the case for sound, a lowering of frequency).
For observers approaching each other, each observes a "blueshift" (or, in the case for sound, a raising of frequency).
In some sense, the Doppler Effect needs the time-dilation factor in order to satisfy the principles of relativity.
Indeed, in the derivation of receding sources and receding receivers,
one gets expressions involving the Galilean-Doppler factor and the time-dilation factor:
$\gamma(1+\beta)=\left(\frac{1}{\sqrt{(1-\beta)(1+\beta)}}\right)(1+\beta)=\sqrt{\frac{1+\beta}{1-\beta}}=k$
and $\frac{1}{\gamma}\left(\frac{1}{1-\beta}\right)=\left(\sqrt{(1-\beta)(1+\beta)}\right)\left(\frac{1}{1-\beta}\right)=\sqrt{\frac{1+\beta}{1-\beta}}=k$.

It might be useful to point out a distinction between time-dilation and the Doppler effect for light.
For two inertial observers Alice and Bob that met at event O,

• time-dilation involves two spacelike-related events,
  say "event P on Alice's worldline" and "event Q on Bob's worldline that Alice says is simultaneous with P"
  (so, $\vec{PQ}$ is a purely-spatial displacement vector according to Alice... it is Minkowski-perpendicular to $\vec{OP}$).
  The time-dilation factor measured by Alice is $\gamma=\frac{OP}{OQ}$.
• Doppler-effect involves two lightlike-related events,
  say "event P on Alice's worldline" and "event S on Bob's worldline which is in the lightlike-future of P"
  (so, $\vec{PS}$ is a future-lightlike displacement vector).
  The Doppler factor measured by Alice is $k=\frac{OS}{OP}$.
  

If these movies were youtube videos, there would be a time indicator rolling at the bottom of the screen, so for example, both Alice and Bob could see that Alice's movie indicates that it starts at 00:00:00 and increments to 00:60:00 at the end. Although Bob can't necessarily "view Alice's clock" , he can see by the video time index that in comparison to his own clock her video is running slow... suggesting that her time is slower relative to his (and likewise his to hers when he sends video to her).

When Alice and Bob approach each other, it looks like Bob is going to see Alice's movie running faster (shorter time), and Bob's movie will be seen by Alice to be running faster... so this is not similar to SR which would maintain that each measure each others clocks running slow.

In the case of approaching, one has a diagram like this [based on reflecting the original diagram from the Insight]:

where I have used a "factor" $\kappa$ (kappa).
So, as you said, Bob would view Alice's T-hour broadcast "sped up", in only $\kappa T$ hours (where $\kappa<1$).
By similar triangles, $\displaystyle\frac{\kappa T}{T}=\frac{kT}{k^2T}$, which implies that $\kappa=\frac{1}{k}$.
Note that since $k=\sqrt{\frac{1+\beta}{1-\beta}}$,
we have $\kappa=\frac{1}{k}=\sqrt{\frac{1-\beta}{1+\beta}}$, which is the original expression for "$k$" with "velocity $-\beta$".
Thus, there's no need to use $\kappa$... "receding and approaching" are handled by $k$.

 

[leader]

The theory of special relativity was derived from a simple fact based on the right triangle as follows :
Imagine a light signal is sent from a point to an observer moving with a velocity "v". This signal will be received by this observer moving with velocity "v" after a time delay with respect to the initial position of this observer that forms the hypotenus of the right triangle on which the velocity of light "c" is the same as the one of the right sides of the right triangle while for the other right side the velocity of the observer is "v". If you multiply these velocities with the same time difference "dt", addition of squares of the two right sides would be greater than the square of the hypotenus that would violate the pytogoran theorem for which it becomes necessary to denominate the time intervals with different indices as : :
(cdt*)^2 = (vdt*)^2 + (cdt)^2 which after a simple algebra becomes dt* = dt / [ 1 - (v/c)^2 ]^1/2

 

[michall]

About forty years before Einstein, Maxwell published equations describing electromagnetism. One solution to the equations was a wave, which turned out to have the properties of light. One weird thing was that the speed of the wave came out the same always. Naturally everyone assumed that the equations weren't quite right and the hunt was on to find the problem.

The next forty years were a bit confusing as no one could find anthing wrong. Experiments that were expected to help (e.g. Michelson and Morley) didn't work as predicted, but did provide some ad hoc patches. Einstein had the insight that if the (apparently daft) prediction that light always travels at the same speed for all inertial observers was correct then he could explain all of the confusion. So he made the assumption.

So he made the assumption
This puts it in the wrong light. Einstein was quite explicit: the constancy of the speed of light is not an assumption (an hypothesis or an empirical generalization). It is a bit of logic that has to do with the only possible way of synchronizing (distant) clocks and measuring time. It is a conceptual necessity, not a fact about the nature of light.

 

[Ibix]

So he made the assumption
This puts it in the wrong light. Einstein was quite explicit: the constancy of the speed of light is not an assumption (an hypothesis or an empirical generalization). It is a bit of logic that has to do with the only possible way of synchronizing (distant) clocks and measuring time. It is a conceptual necessity, not a fact about the nature of light.

I'm not quite sure what you mean by this. In his 1905 paper Einstein calls the constancy of the speed of light a "principle", although it's more often referred to as a "postulate". Either way, it's something he assumed. It's the basis of his reasoning, yes, and it isn't necessary for light to travel at the invariant speed for there to be an invariant speed. But neither fact makes Einstein's postulates anything other than postulates.

 

[Mister T]

It is a bit of logic that has to do with the only possible way of synchronizing (distant) clocks and measuring time.

Taking advantage of the fact that the speed of light is invariant is not the only way to synchronize spatially separated clocks. And it's not necessary for measuring time. Time is a physical quantity. Synchronization of spatially separated clocks is a convention. There are many ways to do it. Some are equivalent and some aren't.

 

[michall]

There is a circularity in the idea of measuring the speed of light: to measure the speed you need synchronized clocks; but you can't get distant clocks synchronized unless you know the speed of light. It's a logical circle. The only way out of it, says Einstein, is to assume (stipulate) that the time it takes light to go from A to B = the time it takes to go from B to A. Now that is how Einstein originally put it, and it makes it sound like an assumption and a matter of convention, which happily, fortuitously, agrees with the actual behavior of light signals. That is not the right way to put it: the language is old fashioned and betrays the thought, which is that the only way to synchronize distant clocks, A and B, is the one-clock method of sending a signal from A to B and back again and taking half of the interval of time and adding it to the start time at A to get the time at B (in accordance with the definition). Some scientists and philosophers take the "1/2" to be a variable, as if it were an arbitrary choice among a range of other possible values, making the definition into a convention. But there are no other possible, ie. empirically determinable, values, because of the logical conundrum above. If there are other way--slow transport is rubbish--they will be logically equivalent to Einstein's, if they are valid, because E. has the concept of simultaneity down pat. It is reflexive, symmetrical, and transitive--the logical form of identity of time--and because it is a one clock measurement of a round-trip, the constancy of the speed of the sigal follows as a logical inference. That's why E. was not surprised at all by the non-outcome of Michelson-Morely.

 

[Mister T]

There is a circularity in the idea of measuring the speed of light: to measure the speed you need synchronized clocks; but you can't get distant clocks synchronized unless you know the speed of light. It's a logical circle.

My desk has a width of 0.500 000 000 m. I shall send a pulse of light from one end to the other and have it reflect back to arrive at the source. The time it took was $\frac{1}{299\ 792\ 458}$ seconds. I therefore measure the speed of light to be 299 792 458 m/s.

There was nothing circular in the logic, and there was no synchronization of clocks. I used only one clock!

 

[Ibix]

There is a circularity in the idea of measuring the speed of light: to measure the speed you need synchronized clocks; but you can't get distant clocks synchronized unless you know the speed of light.

The one-way speed of light, yes. That's why it's the two-way speed that is considered invariant. The one-way speed is conventionally assumed to be equal to the two-way speed, but there's no way to test this.

The only way out of it, says Einstein, is to assume (stipulate) that the time it takes light to go from A to B = the time it takes to go from B to A.

A way out of it. I derived some of the maths for inertial frames assuming otherwise here. It's not difficult, just ugly.

Some scientists and philosophers take the "1/2" to be a variable, as if it were an arbitrary choice among a range of other possible values, making the definition into a convention. But there are no other possible, ie. empirically determinable, values,

Not quite; there are no empirically determinable values for the one way speed of light - that's why it's a convention. Choosing a value other than c is simply choosing a different (non-orthogonal) coordinate system in spacetime. Most people don't do it for inertial frames because it's ugly, unintuitive, makes the maths more complex, and adds nothing over the simple symmetric assumption - but there is nothing preventing it.

and because it is a one clock measurement of a round-trip, the constancy of the speed of the sigal follows as a logical inference.

Why do you think that this doesn't apply to one clock measurements of the round-trip speed of a ball? Or a sound wave? What's special about light that there is a "logical identity" with its behaviour but nothing else?

 

[tom.capizzi]

Any approach I use must use the spacetime diagram
because I think it is difficult to represent the relativity-of-simultaneity using boxcars as "moving frames of reference".

Any approach I use must use radar methods
to motivate measurements and the assignment of coordinates.
I think radar methods are more straightforward than lattices of "clocks" and "rods".
(For inertial motions in special relativity, they are equivalent.
However, for more general motions in special and general relativity, they may differ...
and would require more advanced discussion to address.)

In my opinion, the Bondi k-calculus method (with its emphasis on radar measurements) is the best starting point, especially for algebra-based physics. With the k-calculus methods, the standard textbook formulas are straightforward to derive and fall out naturally.

A related but even less well known approach by Geroch (in his General Relativity from A to B) is also a good starting point. Geroch uses radar methods to emphasize the square-interval and give operational interpretations of the geometry of spacetime (e.g., what simultaneity means to an observer) in both Special Relativity and Galilean Relativity. My AJP article (which inspired the Insight https://www.physicsforums.com/insights/relativity-rotated-graph-paper/ ) was my attempt to combine Bondi's and Geroch's approaches.

From here, I would go on to develop the geometry of Minkowski spacetime, while comparing and contrasting with Euclidean geometry, using the [unappreciated] geometry of Galilean spacetime (e.g., https://www.desmos.com/calculator/ti58l2sair ... play with the E-slider) ...something I call "Spacetime Trigonometry", a large ongoing project with many aspects which generates lots of posters for me at AAPT meetings. (I should really write this up soon... but it would have to be broken into a series of AJP articles.) These are examples of Cayley-Klein geometries, which includes the deSitter spacetimes.. This "unification" can help formalize the numerous analogies mentioned in the literature. In addition, I can develop vector and tensorial methods (algebraically, graphically, and geometrically) in order to make contact with traditional intermediate and advanced presentations of relativity.

Are you aware that the Bondi approach is a half-hearted attempt at eigenvector decomposition? The k-factor in Bondi k-calculus is, in fact, an eigenvalue of the Lorentz matrix. The radar measurements are another feature of eigenvector analysis, because the eigenvectors of the 2x2 Lorentz matrix are ct+r=0 and ct-r = 0. Note to PF moderators. That is not my opinion, that's a fact. Look it up. Other than those lines are identical to the worldlines of photons, and are built-in radar (or lidar) measurements, I can't say more. PF moderators think that mathematical analysis is my "personal opinion", and gave me non-expiring demerits for posting math. I would close my account, but their privacy policy won't let me. So I will continue to post inconvenient mathematical truth until they ban me for life, which I will accept as a badge of honor. This site is too conservative.

 

[robphy]

Are you aware that the Bondi approach is a half-hearted attempt at eigenvector decomposition? The k-factor in Bondi k-calculus is, in fact, an eigenvalue of the Lorentz matrix. The radar measurements are another feature of eigenvector analysis, because the eigenvectors of the 2x2 Lorentz matrix are ct+r=0 and ct-r = 0. Note to PF moderators. That is not my opinion, that's a fact. Look it up. Other than those lines are identical to the worldlines of photons, and are built-in radar (or lidar) measurements, I can't say more. PF moderators think that mathematical analysis is my "personal opinion", and gave me non-expiring demerits for posting math. I would close my account, but their privacy policy won't let me. So I will continue to post inconvenient mathematical truth until they ban me for life, which I will accept as a badge of honor. This site is too conservative.

I'm not sure what you mean by half-hearted...
but yes, I am aware. From my Insight articles:

These are related to “light-cone coordinates“, the coordinates in the eigenbasis of the Lorentz Transformation. In my convention, Δu=Δt+Δxc and Δv=Δt−Δxc. So, these equations can be rewritten as Δu=k^2T and Δv=T, The k-factor is an eigenvalue of the Lorentz Transformation

https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/

The Doppler-Bondi k-factor is the eigenvalue along the future-forward direction, which stretches the light-clock diamond in that direction by a factor k. The reciprocal $k^{−1}$ is the eigenvalue along the future-backward direction, which shrinks the light-clock diamond by a factor of k. Note the light-clock diamond area is preserved. This is the basis of my “Relativity on Rotated Graph Paper” Insight.

Source https://www.physicsforums.com/insights/relativity-variables-velocity-doppler-bondi-k-rapidity/

 

[vanhees71]

Of course the light-cone vectors are eigenvectors of the Lorentz transformation. The pity with Bondi calculus is that it's elegance is hidden under all these diagrams. Algebra is often much easier to grasp. Here's the simple algebra:

https://www.physicsforums.com/threa...ept-from-first-principles.984645/post-6547265

Now indeed the light-cone basis vectors in terms of the Minkowski-orthgonal basis are $\ell_1=(1,-1)$ and $\ell_2=(1,1)$. These are indeed eigenvectors of eigenvalues $k$ and $1/k$ respectively. In terms of the rapidity $\eta$ defined by $v=\tanh \eta$ you have $k=\exp \eta$.