Understanding Proper Length in Special Relativity

[Beelzebro]

I'm confused about what exactly this is, and wikipedia hasn't really helped.

It says here the proper length is the length of an object in it's rest frame.

Yet it says here that "the proper length between two spacelike-separated events is the distance between the two events, as measured in an inertial frame of reference in which the events are simultaneous".

If for example a train goes past a platform at a relativistic speed and an observer on the platform sees two strikes of lightning hit each end of the train simultaneously, and they're 50m apart, then what is the proper length of the train? Is it 50m because the two events are simultaneous in this frame? Or is it 50m multiplied by the lorentz factor, because this would be the value in the trains rest frame? (since for the platform observer the train appears contracted).

I do not see how the two definitions are consistent and I'm not sure which one I'm supposed to use.

 

[Bill_K]

The proper distance, or interval, between the two lightning strikes is, as your second reference says, the distance as measured in a frame in which the events are simultaneous. 50 m in this case.

The proper length of the train is measured in the object's own rest frame, 50 m * gamma in this case.

 

[PAllen]

To avoid confusion, I've come to prefer the less common terminology:

-rest length (of an object)

- proper distance (between two events)

 

[HallsofIvy]

I'm confused about what exactly this is, and wikipedia hasn't really helped.

It says here the proper length is the length of an object in it's rest frame.

Yet it says here that "the proper length between two spacelike-separated events is the distance between the two events, as measured in an inertial frame of reference in which the events are simultaneous".

This is the same as saying "in the rest frame of an observer who see both events as simultaneous". As others have said, it is the difference between "length of an object", in which time is irrelevant, and "distance between events" in which time is relevant. It is essentially saying "find a frame in which the two events happen at the same time and measure the length of an object extending from the point where one event occurred to the point where the other event occured.

If for example a train goes past a platform at a relativistic speed and an observer on the platform sees two strikes of lightning hit each end of the train simultaneously, and they're 50m apart, then what is the proper length of the train? Is it 50m because the two events are simultaneous in this frame? Or is it 50m multiplied by the lorentz factor, because this would be the value in the trains rest frame? (since for the platform observer the train appears contracted).

The distance between the two events is should not be the same as the "proper length" of the train. The only reason you think these definitions are not consistent is that you think they are. Yes, the observer on the platform sees the length of the train being the same as the distance between those two points. But that is NOT the "proper length" of the train because the observer is not in the rest frame of the train.

I do not see how the two definitions are consistent and I'm not sure which one I'm supposed to use.

What are you measuring? The proper length of the train or the distance between lightning strikes? If the proper length of the train, board the train and measure it while you are moving with it. If the distance between lightning strikes, measure on the platform.

 

[Naty1]

I do not see how the two definitions are consistent...

Lucky you, posts from three experts...good explanations...but so does your first Wikipedia:

First it is necessary to carefully consider the methods for measuring the lengths of resting and moving objects, where "object" simply means a distance with endpoints that are always mutually at rest, i.e., that are at rest in the same inertial frame of reference. If the relative velocity between an observer (or his measuring instruments) and the observed object is zero, then the proper length of the object can simply be determined by directly superposing a measuring rod. However, if the relative velocity > 0, then one can proceed as follows...Thus the definition of simultaneity is crucial for measuring the length of moving objects. In Newtonian mechanics, simultaneity is absolute and therefore and are always equal.

You have to really read this stuff slowly and carefully...as least I always have had to...it is subtle. I know I sure had trouble figuring that out when I was starting.

I don't believe I thought about it carefully enough until I started with cosmoligcal distances and where 'proper distance' requires yet more thinking since actual distances are changing due to the expansion of space with time.

Anyway, when you take a typical LENGTH measure in everyday life, you measure a LOCAL length, such as a 6 ft long piece of wood. What are you REALLY doing: Einstein must have thought about examples like that REALLY carefully. What is not obvious is that both you and the wood are 'moving' together: that is, there is no relativistic velocity between you, no relative acceleration...you are in the same frame of reference...and you take the measure between the two endpoints 'simultaneously'. That is, the opposite end of the ruler is on a fixed stationary point as you read off a length measure at the other end..…it’s a ‘simultaneous measure’. [If I ever knew that way back in college I sure had forgotten since then!]

In everyday situations, if you measured that wood from some finite distance away, say using an instrument, you'd have to adjust the measure for the distance between you and the wood, but you'd likely not worry about time variations between you and either end of the wood. [The finite speed of light is a negligible factor.]

What can quickly become confusing in relativity is that when observational distances are involved, or relative motion, space and time morph into each other...neither is fixed and static as in the simple case...space and time become dynamic. You have to adjust observational measurements among different frames of reference in order to make comparisons . A measure from a ‘distance’ [space,time,motion] becomes one of space and TIME.

 

[Beelzebro]

Ok, I think you guys have cleared it up for me, thanks so much :D :D

I was confused, like you have pointed out, with the distinction between measuring the distance between the two lightning strikes, and measuring the length of the train.

Thanks again :)