Lorentz transformations, time intervals and lengths

[bernhard.rothenstein]

(have a square-root: √ and a gamma: γ )

oh i see …

you're saying that in the equations

dx = γ(dx' + vdt')
dt = γ(dt' + vdx'/c²)​

x and vt both have dimensions of length, so as a matter of English is it proper to call them both lengths?

in other words, just as x is naturally a "proper" length, is vt also a "proper" length?

My answer would be that, to familiarise students with "space-time" and the interchangeability of space and time, and particularly the rotational nature of a Lorentz boost (which obviously requires like to be rotated onto like),

it's best to use ct and v/c …

dx = γ(dx' + (v/c)d(ct'))
d(ct) = γ((d(ct') + (v/c)dx')​

… in other words, to present ct as a length (rather than vt), and v/c as an ordinary number …

and indeed to avoid using a "naked" vt at all.

Presenting the Lorentz-Einstein transformations as proposed above and taking into account the way in which they are measured dx, dx', d(ct), d(ct') represent lengths of objects at rest in I and in I' respectively we could say that they are proper lengths measured in I and in I' respectively. In what concerns t and t', taking into account the way in which they are measured represent coordinate time separations. Taking into account that V and c are measured as a quotient between a proper length and a coordinates time separation, V/c is a number. Should a learner know all that?
Please tell me if I deserve an optimistic smily?

 

[tiny-tim]

Please tell me if I deserve an optimistic smily?

you can always award yourself a smilie!

Presenting the Lorentz-Einstein transformations as proposed above and taking into account the way in which they are measured dx, dx', d(ct), d(ct') represent lengths of objects at rest in I and in I' respectively

we could say that they are proper lengths measured in I and in I' respectively.

But "proper time" is the "own-time" measured on a clock (stationary or otherwise) … this is standard terminology.

In what sense is dx a "proper length" in the same way?

Surely this over-use of the word "proper" will just confuse students?

In what concerns t and t', taking into account the way in which they are measured represent coordinate time separations. Taking into account that V and c are measured as a quotient between a proper length and a coordinates time separation, V/c is a number. Should a learner know all that?

A learner should certainly understand that v/c is a number, and probably that its inverse tanh is what wikipedia calls "rapidity", and is additive (in one dimension).

But again won't talk of "time separation" confused students, by using "separation" which has a distinct meaning which is already standard terminology?

 

[atyy]

I'm quoting from memory, so not sure this is right. Anyway, I think 't Hooft says he deliberately uses different notations since students should get used to it (I read that as students should get used to being confused :tongue2:). Also I think Wald defines proper time as the "length" of a timelike curve, and proper length as the "length" of a spacelike curve, with "lengths" of curves that switch from timelike to spacelike as being undefined. But I think Wald also has some unconventional definitions about Christoffel symbol-like things being tensors.

Edit: There has to be a minus sign to go with one of the "lengths".

 

[bernhard.rothenstein]

you can always award yourself a smilie!


But "proper time" is the "own-time" measured on a clock (stationary or otherwise) … this is standard terminology.

In what sense is dx a "proper length" in the same way?

Surely this over-use of the word "proper" will just confuse students?


A learner should certainly understand that v/c is a number, and probably that its inverse tanh is what wikipedia calls "rapidity", and is additive (in one dimension).

But again won't talk of "time separation" confused students, by using "separation" which has a distinct meaning which is already standard terminology?

I leave the thread learning the following facts:
Even natives do not agree when it is about giving names to physical quantities measured following a given method. So I think that it is better to avoid additive names to a physical quantity in order to show the way in which it was measured. So I will use teaching the following strategy: Speaking about a physical quantity it is advisable to define it, to specify the observer who measures it, the point in space and the time when the measurement is performed and the device used to measure it.
Speaking about time I would use the term elapsed time between two events which can be measured using a single clock that is present at both events or by a pair of synchronized clocks, one present at one event, and the other present at the other event.
Speaking about the length of an object in a given inertial reference frame it can be measured simultaneously detecting the space coordinates of its ends and taking their difference. In the particular case when the object is in a state of rest in a given reference frame the condition of simultaneity is not compulsory. Doing so I confuse the students?
Finding names for the mentioned cases, it is illusory to think that they will be accepted by large communities of physicists. See the case of mass.
Smiles please.

 

[tiny-tim]

:tongue2: :rofl: ​

and of course …

​

 

[bernhard.rothenstein]

:tongue2: :rofl: ​

and of course …

​

Please tell me if "proper time SPAN" and coordinate time SPAN" sounds well?

 

[tiny-tim]

Please tell me if "proper time SPAN" and coordinate time SPAN" sounds well?

hmm … to me, it sounds repetitious …

we say "a distance of 3 miles", not "a distance span of 3 miles" …

what does the word "span" add to "proper time" or "coordinate time"?

(i agree we do sometimes say "time-span", with a hypen, as in "over a time-span of centuries" … but that's really only where, in ordinary English, it we might be misunderstood as referring to a "point in time")