Lorentz transformations, time intervals and lengths

[tiny-tim]

My initial question, giving rise to so much noise, was:
Present the Lorentz transformations as
(1-bb)^1/2(x-0)=(x'-0)+V(t'-0)
Being new in English terminology, I asked how would you call
(x-0), (x'-0) and V(t'-0) taking into account that all have the physical dimension of length.
Teaching special relativity is it necessary to make a distinction between them?

(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.

 

[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.

Nice to meet you on the forum. I think that is the elegant way to share knowledge when somebody solicitates it.
From the way in which you present the Lorentz transformations (I will use them in my work)
I think that we could conclude that dx, dx' and d(ct') are proper lengths or they are not?
If we make a distinction between the different types of time interval it is not clear fro me what kind of time intervals are dt and dt'?
With kind regards

 

[tiny-tim]

hi bernhard!

… I think that we could conclude that dx, dx' and d(ct') are proper lengths or they are not?

I really don't think you should use the word "proper" …

a length is a length, and you don't have to praise it as being a "genuine" length or a "proper" length …

and (ct) is also a length … just a length!

Not only is it unnecessary, but it also risks confusion with the "proper length", tau, along a curve.

I think students should be told "spacetime has four dimensions, x y z and ct, and they're all lengths"

If we make a distinction between the different types of time interval it is not clear fro me what kind of time intervals are dt and dt'?

I honestly don't think one should talk about t (or dt) in terms of intervals … after all, they don't have lengths, do they?

And what good is an interval without a length? ​

 

[bernhard.rothenstein]

Thanks. I quote from a textbook from which I have learned a lot:
"ds=dt/g
this very useful equation links the spacetime interval ds measured by an inertial clock present at two events with the coordinate time separation dt between those events in some inertial frame and the speed v of the clock as measured in the same inertial frame".
When applying this equation, it is important to remember two things. First of all, coordinate time dt and spacetime interval ds represent the time interval between two events measured in two fundamental different ways. The coordinate time between events is measured with a pair of synchronized clocks in an inertial frame, while the spacetime interval is measured by an inertial clock present at both events."
Is all that an out of fashion way to teach special relativity? Is there some danger of missinterpretation?
With thanks and respect

 

[tiny-tim]

Thanks. I quote from a textbook from which I have learned a lot:
"ds=dt/g

(what happened to that γ i gave you? )

Of course, that's with units in which c = 1 …

so speed is a dimensionless number, and length and time are dimensionally the same.

… When applying this equation, it is important to remember two things. First of all, coordinate time dt and spacetime interval ds represent the time interval between two events measured in two fundamental different ways. The coordinate time between events is measured with a pair of synchronized clocks in an inertial frame, while the spacetime interval is measured by an inertial clock present at both events."
Is all that an out of fashion way to teach special relativity? Is there some danger of missinterpretation?

Personally, I dislike the word "interval".

"Interval" in English isn't a measurement, it's a one-dimensional region.

I'd use the standard term "separation" … "the separation is measured by an inertial clock present at both events."

Apart from that, the explanation seems fine.

(Though it doesn't work where the separation is negative … )​

 

[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 ). 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]

​

and of course …

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[bernhard.rothenstein]

​

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")