Proper Length and Time Intervals in the Lorentz Transformations

[bernhard.rothenstein]

Please tell me if the following statement is correct:
In
cc(dt)^2-(dx)^2=cc(dt')^2-(dx')^2
dx and dx' represent proper length dt and (dt') representing non-proper time intervals.

 

[Fredrik]

It would be easier to read your posts if you used LaTeX or at least the "sup" vbtag, like this: dx².

The integral of \sqrt{|-dt^2+dx^2|} (units such that c=1) along a curve is called "proper time" if the curve is timelike and "proper length" if the curve is spacelike.

 

[bernhard.rothenstein]

It would be easier to read your posts if you used LaTeX or at least the "sup" vbtag, like this: dx².

The integral of \sqrt{|-dt^2+dx^2|} (units such that c=1) along a curve is called "proper time" if the curve is timelike and "proper length" if the curve is spacelike.

Thanks. Is there a simpler way to answer my question?

 

[Gokul43201]

...dx and dx' represent proper length...

There should exist only one frame where you measure the proper length of an object, no?

 

[Dale]

Please tell me if the following statement is correct:
In
cc(dt)^2-(dx)^2=cc(dt')^2-(dx')^2
dx and dx' represent proper length dt and (dt') representing non-proper time intervals.

No, dx dx' dt dt' are all coordinate intervals.

 

[JesseM]

Please tell me if the following statement is correct:
In
cc(dt)^2-(dx)^2=cc(dt')^2-(dx')^2
dx and dx' represent proper length dt and (dt') representing non-proper time intervals.

It's correct if you're talking about two events where the spatial separation between them is dx and the time interval between them is dt in your first inertial coordinate system, while in your second inertial coordinate system the spatial separation is dx' and the time interval is dt'. In SR this will work for two events that are arbitrarily far apart, in a curved spacetime of GR it only works locally (basically considering two events that are infinitesimally close together). Also, note that if you are using three spatial dimensions x, y and z as opposed to just one, then the equation would have to be written as:

c^2 * (dt)^2 - (dx)^2 - (dy)^2 - (dz)^2 = c^2 * (dt')^2 - (dx')^2 - (dy')^2 - (dz')^2

of course, since by the pythagorean theorem we know the total distance dL between two spatial coordinates must be equal to sqrt(dx^2 + dy^2 + dz^2), the above equation reduces to:

c^2 * (dt)^2 - (dL)^2 = c^2 * (dt')^2 - (dL')^2

...where dL is the total spatial separation between the events in the first coordinate system and dL' is the total spatial separation between the events in the second coordinate system.

 

[Fredrik]

It's correct if you're talking about two events where the spatial separation between them is dx and the time interval between them is dt in your first inertial coordinate system, while in your second inertial coordinate system the spatial separation is dx' and the time interval is dt'.

I'm pretty sure that's what he meant, but it's not a great idea to call dx and dx' proper lengths. Proper length is a coordinate-independent quantity that's only defined along spacelike curves.

 

[Fredrik]

Thanks. Is there a simpler way to answer my question?

If you want a yes/no answer rather than an explanation of what proper length is, then there is a simple answer: The dx, dx', dt and dt' are all coordinate-dependent quantities, but proper length is something coordinate-independent.

 

[robphy]

An interval is measurement involving [the spacetime displacement between] two events.

Note that "proper length" is more correctly defined as the "distance" between two parallel timelike-lines on a spacetime diagram. So, some additional structure needs to be specified to make any reference to a "proper length".

It is [often] helpful to consider the Euclidean analogue of the problem.
As others have mentioned, dx,dx',dt,dt' are (in general) coordinate-dependent quantities.
They are essentially the legs of some "right"-triangle where the hypotenuse joins the two events. The freedom to choose among these right-triangles is essentially the coordinate-dependence.

 

[bernhard.rothenstein]

An interval is measurement involving [the spacetime displacement between] two events.

Note that "proper length" is more correctly defined as the "distance" between two parallel timelike-lines on a spacetime diagram. So, some additional structure needs to be specified to make any reference to a "proper length".

It is [often] helpful to consider the Euclidean analogue of the problem.
As others have mentioned, dx,dx',dt,dt' are (in general) coordinate-dependent quantities.
They are essentially the legs of some "right"-triangle where the hypotenuse joins the two events. The freedom to choose among these right-triangles is essentially the coordinate-dependence.

Thanks to all. I rephrase my question. Consider the Lorentz transformations
\Deltax=\gamma[\Deltax'+V\Deltat']
It is correct to say that \Deltax and \Deltax' are proper lengths
\Deltat and \Deltat' representing non-proper time intervals.
Please consider the concept of proper length and non-proper time interval as defined
say in
Thomas A. Moore A Travelers Guide to Spacetime McGraw-Hill,Inc 1995 pp. 126 and 46.
V in the Lorentz transformation is measured as a quotient between a proper length and a nonproper time interval.

 

[JesseM]

Thanks to all. I rephrase my question. Consider the Lorentz transformations
\Delta x=\gamma[\Delta x'+V\Delta t']
It is correct to say that \Delta x and \Delta x' are proper lengths
\Delta t and \Delta t' representing non-proper time intervals.
Please consider the concept of proper length and non-proper time interval as defined
say in
Thomas A. Moore A Travelers Guide to Spacetime McGraw-Hill,Inc 1995 pp. 126 and 46.
V in the Lorentz transformation is measured as a quotient between a proper length and a nonproper time interval.

Your LaTeX code wasn't working, so I fixed it in the above quote. Anyway, I don't know of any definition of "proper length" and "proper time" such that your statement would be true. Normally "proper length" refers to the length of an object in its own rest frame, but you are just talking about the coordinate distance between two events. Perhaps you could quote the definition of proper length given in "A Travelers Guide to Spacetime" since I don't have a copy of that book?

 

[bernhard.rothenstein]

space time interval question

Your LaTeX code wasn't working, so I fixed it in the above quote. Anyway, I don't know of any definition of "proper length" and "proper time" such that your statement would be true. Normally "proper length" refers to the length of an object in its own rest frame, but you are just talking about the coordinate distance between two events. Perhaps you could quote the definition of proper length given in "A Travelers Guide to Spacetime" since I don't have a copy of that book?

I quote from Moore:
"We might say that to measure the "true" length of an object in space-time we should measure its length in the inertial frame in which it is at rest. This "true length" (sometimes called proper length) of the object will be longer than the value measured in any other inertial reference frame. For clarity's sake, let us refer to this length as the rest length of the object."
Are dx and dx' in the LT proper length in accordance with that definition?
I quote again from Moore:
"The time between two events measured by any clock present at both events is called a proper time. A proper time measured by a given clock is an absolute quantity independent of reference frame."
Are dt and dt' in the LT proper time intervals or they are non-proper or distorted time intervals?
Is V in LT measured as a quotient between a proper length and a nonproper time interval?
Thanks for your help.

 

[Dale]

In any SR inertial frame, if two events have a spatial coordinate separation of dx and a temporal coordinate separation of dt then:
1a) if the events are timelike separated the proper time between them is sqrt(dt²-dx²/c²)
1b) |dt| is the proper time between the events iff dx=0
2a) if the events are spacelike separated the proper distance between them is sqrt(dx²-c²dt²))
2b) |dx| is the proper distance between the events iff dt=0

 

[country boy]

I quote from Moore:
"We might say that to measure the "true" length of an object in space-time we should measure its length in the inertial frame in which it is at rest. This "true length" (sometimes called proper length) of the object will be longer than the value measured in any other inertial reference frame. ...

There is an ambiguity in Moore's use of "proper" in "proper length." Unlike "proper time," which is the difference in time between two events (the measurements of a clock at rest), the length of a rod is not the difference between two unique events. If I refer to a rod at rest and define "proper length" to be the distance between the two ends of the rod as measured simultaneously, those measurements are indeed two unique events. [The measured distance then corresponds to the "proper distance" as defined above by DaleSpam.] However, the confusion comes when this "proper distance" is used as Moore's "proper length," determined by measurements performed at arbitrary times. The length of the rod at rest is not a "proper" distance in the same sense as "proper" time.

So, what about cc(dt)^2-(dx)^2=cc(dt')^2-(dx')^2 ? Both dx and dx' can be equal to a proper length by Moore's definition. If they refer to the same rod then dx=dx', and the equation can only be true if dt=dt'. If dx and dx' are equal to the proper lengths of two different rods then dx may not equal dx', but the equation can still be true for special combinations of dt and dt'. But if the equation refers to a measurement of DaleSpam's "proper distance" between two events, then |dx|=|dx'|=ds and dt=dt'=0.

 

[bernhard.rothenstein]

space time interval question

There is an ambiguity in Moore's use of "proper" in "proper length." Unlike "proper time," which is the difference in time between two events (the measurements of a clock at rest), the length of a rod is not the difference between two unique events. If I refer to a rod at rest and define "proper length" to be the distance between the two ends of the rod as measured simultaneously, those measurements are indeed two unique events. [The measured distance then corresponds to the "proper distance" as defined above by DaleSpam.] However, the confusion comes when this "proper distance" is used as Moore's "proper length," determined by measurements performed at arbitrary times. The length of the rod at rest is not a "proper" distance in the same sense as "proper" time.

So, what about cc(dt)^2-(dx)^2=cc(dt')^2-(dx')^2 ? Both dx and dx' can be equal to a proper length by Moore's definition. If they refer to the same rod then dx=dx', and the equation can only be true if dt=dt'. If dx and dx' are equal to the proper lengths of two different rods then dx may not equal dx', but the equation can still be true for special combinations of dt and dt'. But if the equation refers to a measurement of DaleSpam's "proper distance" between two events, then |dx|=|dx'|=ds and dt=dt'=0.

Thanks. Let consider the Lorentz transformations for the space-time coordinates of the same event. They relate dx,dx',dt and dt'. As detected from I and I' respectively dx and dx' represent proper lengths and theit magnitude does not depend on the way in which the corresponding observers measure it. dt and dt' measured from I and I' respectively are nonproper time intervals measured in Einstein's approach being measured as a difference between the readings of two distant clocks synchronized in accoprdance with the procedure he proposed. The speed V in the LT is measured as a quotient between a proper length and a nonproper time interval.
Other synchronization procedures and other measurement conventions lead to changes in the physical meaning of the physical quantities involved in the LT. So a time transformation could relate two nonproper time intervals, a poper and a nonproper time interval or even two proper time intervals. So a "everyday clock synchronization procedured' proposed by Leubner leads to absolute simultaneity and length dilation. Such situations can be found in the case of a photographic detection of moving rods and clocks. An interesting situation is presented when a convention is made that the moving observer measures distances on a chart devised on Earth and time using his wrist watch. In all cases the Authors reinvent Einstein's special relativity and space-time diagrams addapted to the imposed conditions. What I try to do is to show that the LT as presented by Einstein account for all the possible and faisible scenarios.
As a lonely physicist I would highly appreciate your opinion telling me where I am wrong in the lines above.
Thanks in advance. Please take into account that English is not my first language and that I prepare the paper for readers who know SR from standard texts.

 

[Dale]

Other synchronization procedures and other measurement conventions lead to changes in the physical meaning of the physical quantities involved in the LT. So a time transformation could relate two nonproper time intervals, a poper and a nonproper time interval or even two proper time intervals. So a "everyday clock synchronization procedured' proposed by Leubner leads to absolute simultaneity and length dilation.

It is very unproductive to use a standard formula, standard variables, and standard terminology in your initial post and then wait until much later to mention the fact that you are using non-standard definitions of more fundamental things like space and time. If you want to use anything other than standard definitions of things you should point that out explicitly in your original post.

 

[country boy]

... Let consider the Lorentz transformations for the space-time coordinates of the same event. They relate dx,dx',dt and dt'. As detected from I and I' respectively dx and dx' represent proper lengths ...

As I see it, the problem is still with the meaning of "proper length." Any measurement of distance using a stationary ruler can be called a proper length. Likewise, any measurement of elapsed time using a stationary clock can be called a "proper time." But these measurements, if performed at different times or places from the events themselves will not be the space-time coordinate differences that appear in the Lorentz transformations.