Relativity of Simultaneity and Length Contraction

[Orodruin]

It's been 30 years since undergrad course covering SR for me, and I'm not trying to insist on my definition of "fundamental", but that is the word Morin uses for these three effects.

He is missing a ”-ly confusing”

Jokes aside, students arriving in my intro SR course have soooo many misconceptions regarding relativity from whatever they learned from some modern physics course that typically emphasize time dilation and length contraction. They try to apply it blindly without really understanding relativity of simultaneity nor the limitations of the expressions. Driving these misconceptions out is half the work and still you see some students get it wrong on the exam …

 

[Nugatory]

typically emphasize time dilation and length contraction. They try to apply it blindly without really understanding relativity of simultaneity

The coin didn’t really drop for me until I learned to think of time dilation and length contraction as manifestations of relativity of simultaneity:
- derive the length contraction formula by working out where the two ends of a rod are at the same time.
- derive the time dilation formula by working out what the other clock reads at the same time that my clock reads T.

 

[Orodruin]

@Orodruin this might be a good spot to link to any material you have published using your approach. It is the second page of an ongoing thread where it is pertinent, so it isn’t really spammy.

I have a set of extensive lecture notes, but nothing really published. They are a number of years old as well so I should probably revise them …

I start by deriving the LTs in analogy with rotations in 2D, then discuss some geometry. Length contraction and time dilation are relegated to section 4 and I discuss them mainly because if I don’t students will still use them as they have ”learned” it before. The discussion is necessary to try to set things straight …

 

[Orodruin]

The coin didn’t really drop for me until I learned to think of time dilation and length contraction as manifestations of relativity of simultaneity:
- derive the length contraction formula by working out where the two ends of a rod are at the same time.
- derive the time dilation formula by working out what the other clock reads at the same time that my clock reads T.

I’d even go so far as to argue you have not really understood them until you realize what their Euclidean analogies are … once there it is a an ”aha moment” and a couple of minutes later it all seems trivial.

 

[Nugatory]

I’d even go so far as to argue you have not really understood them until you realize what their Euclidean analogies are … once there it is a an ”aha moment” and a couple of minutes later it all seems trivial.

Yep, that’s the coin-dropping moment.

 

[curiousburke]

The coin didn’t really drop for me until I learned to think of time dilation and length contraction as manifestations of relativity of simultaneity:
- derive the length contraction formula by working out where the two ends of a rod are at the same time.
- derive the time dilation formula by working out what the other clock reads at the same time that my clock reads T.

Yes, this is very similar to what I was trying get at.

I'll have to work specifically through those as you said.

 

[robphy]

I’d even go so far as to argue you have not really understood them until you realize what their Euclidean analogies are … once there it is a an ”aha moment” and a couple of minutes later it all seems trivial.

I'd go further and develop a trigonometric approach,
especially since trigonometry is probably more familiar.
The time-dilation $\gamma$-factor is essentially the hyperbolic-cosine function of rapidity,
interpreted as "adjacent over hypotenuse".
The dimensionless-velocity $\beta$-factor is the hyperbolic-tangent function of rapidity.
The Doppler $k$-factor is the exponential function.

 

[curiousburke]

I’d even go so far as to argue you have not really understood them until you realize what their Euclidean analogies are … once there it is a an ”aha moment” and a couple of minutes later it all seems trivial.

Do you mean Euclidean analogs of length contraction and time dilation?

 

[curiousburke]

I'd go further and develop a trigonometric approach,
especially since trigonometry is probably more familiar.
The time-dilation $\gamma$-factor is essentially the hyperbolic-cosine function of rapidity,
interpreted as "adjacent over hypotenuse".
The dimensionless-velocity $\beta$-factor is the hyperbolic-tangent function of rapidity.
The Doppler $k$-factor is the exponential function.

Now you're just making stuff up:)

 

[Orodruin]

I'd go further and develop a trigonometric approach,
especially since trigonometry is probably more familiar.
The time-dilation $\gamma$-factor is essentially the hyperbolic-cosine function of rapidity,
interpreted as "adjacent over hypotenuse".
The dimensionless-velocity $\beta$-factor is the hyperbolic-tangent function of rapidity.
The Doppler $k$-factor is the exponential function.

That’s not going further though. It is doing exactly what I do. (Although I never introduce $\beta$ as (almost) the first thing I do is to let $c=1$ by convention.)

 

[Orodruin]

Do you mean Euclidean analogs of length contraction and time dilation?

Yes. They are just geometrical effects in Minkowski space. Relating sides of a triangle if you will. No stranger than relating the hypothenuse to one of the sides with the cosine.

 

[Nugatory]

Now you're just making stuff up:)

He is not....
and yes, I see your smiley so I know that you're not being completely serious but I am going to respond more seriously....

All of high-school trig is just working through the consequences of the Pythagorean theorem $\Delta S^2=\Delta x^2+\Delta y^2$ (include $\Delta z^2$ if you must, but we can orient our coordinates axes so that $\Delta z$ is zero, which is how high school trig textbooks can print their diagrams on two-dimensional pages).

Minkowski space works the same way except that we start with $\Delta S^2=\Delta x^2-\Delta t^2$ (after we've chosen our coordinates such that $\Delta y$ and $\Delta z$ are both zero). That minus sign is the difference between Euclidean and Minkowski space, and it doesn't break the analogy.

 

[robphy]

Now you're just making stuff up:)

:)
See my old posts
https://www.physicsforums.com/threads/trying-to-understand-space-time-diagrams.1045805/post-6803852
and
https://www.physicsforums.com/threads/trying-to-understand-space-time-diagrams.1045805/post-6803784

 

[Orodruin]

:)
See my old posts
https://www.physicsforums.com/threads/trying-to-understand-space-time-diagrams.1045805/post-6803852
and
https://www.physicsforums.com/threads/trying-to-understand-space-time-diagrams.1045805/post-6803784

This reminds me of the old sector area vs arc length usage for hyperbolic trig functions. You can use both equally well of course as long as you use the Minkowski metric for the arc length. The area form is the same in Minkowski and Euclidean space though so it is somewhat more universal to use the sector area. It is also where the nomenclature for the inverse functions comes from (eg, it is arsinh for area, not arcsinh for arc). Arguably you could also use the unit circle area for regular angles as well so using arsin instead of arcsin would not be too much of a stretch. Of course you can always just use asin and sweep the entire debate under the carpet.

 

[DrGreg]

Of course you can always just use asin and sweep the entire debate under the carpet.

I was brought up to use $\sin^{-1}$ etc. I think that might be a U.K. vs U.S. thing.

 

[Orodruin]

I was brought up to use $\sin^{-1}$ etc. I think that might be a U.K. vs U.S. thing.

My "beef" with that is the risk of confusion between csc and asin. After all, we write $\sin^2(x)$ implying the square of the sine so interpreting $\sin^{-1}(x)$ as $1/\sin(x) = \csc(x)$ is not a long shot.

 

[DrGreg]

My "beef" with that is the risk of confusion between csc and asin. After all, we write $\sin^2(x)$ implying the square of the sine so interpreting $\sin^{-1}(x)$ as $1/\sin(x) = \csc(x)$ is not a long shot.

I agree, there is an inconsistency in the notation.

 

[robphy]

In discussions with physics teachers, rapidity is (sadly) sometimes considered an advanced topic.
(See https://www.physicsforums.com/threa...etime-physics-by-wheeler.1004722/post-6513631 )

In typical problems, one doesn't need the value of the rapidity.
One can get by with ratios:
"$\beta=\tanh\theta=\frac{OPP}{ADJ}=\rm (slope)$" and
"$\gamma=\cosh\theta=\frac{ADJ}{HYP}$" and
"$\beta\gamma=\sinh\theta=\frac{OPP}{HYP}$".
One just needs to know how to recognize Minkowski-right-triangles in a spacetime diagram,
then generalize the familiar formulas for "trig-functions as ratios of right-triangle-legs".

 

[Orodruin]

In discussions with physics teachers, rapidity is (sadly) sometimes considered an advanced topic.

Some people have little respect for properly using an additive parameter for their continuous one-parameter groups … sigh

 

[robphy]

Some people have little respect for properly using an additive parameter for their continuous one-parameter groups … sigh

$\xi$ ?

Some people prefer to use a parameter, which is actually non-additive,
but treat it as additive [due to inappropriate extrapolation].

$v$