Derivation of Proper Time and Proper Space

Click For Summary

Discussion Overview

The discussion revolves around the derivation of formulas for proper time and proper space (length) using the Lorentz Transformation equations. Participants explore the definitions and relationships between proper time, coordinate time, proper length, and coordinate length, with a focus on the implications of these derivations in the context of special relativity.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents a derivation for proper time and proper space using the Lorentz Transformation equations, suggesting that proper time is given by τ = t * sqrt(1 - v²/c²) and proper space by σ = x * sqrt(1 - v²/c²).
  • Another participant questions the clarity of the definitions of sigma (σ) and tau (τ), arguing that the relationship between proper and coordinate lengths and times is not adequately addressed.
  • Some participants assert that proper length should be the longest length of an object, challenging the derived formulas that suggest otherwise.
  • A participant emphasizes the importance of distinguishing between different reference frames and correctly applying definitions of proper time and proper length, stating that proper time is defined as the time between two events occurring at the same location in an inertial frame.
  • There is a discussion about the implications of measuring lengths and times in different frames, with one participant arguing that the derived equations incorrectly mix frames.
  • Several participants engage in clarifying the definitions and assumptions underlying the derivations, with some providing corrections and alternative perspectives on the proper definitions.

Areas of Agreement / Disagreement

Participants express disagreement regarding the correctness of the derivations presented, particularly in relation to the definitions of proper time and proper length. There is no consensus on the validity of the initial derivations, and multiple competing views remain regarding the interpretation of the results.

Contextual Notes

Some participants note that the assumptions made in the derivations may not align with standard definitions in special relativity, particularly regarding the conditions under which proper time and proper length are defined. There is also mention of the need for clarity in notation and the implications of simultaneity in different reference frames.

Who May Find This Useful

This discussion may be of interest to students and enthusiasts of physics, particularly those studying special relativity and the Lorentz Transformation, as well as individuals seeking to understand the nuances of proper time and proper length in different inertial frames.

AdVen
Messages
71
Reaction score
0
I have tried to derive the formula for the proper time from the two equations of the Lorentz Transformation. The formula is as follows (see Wikipedia: http://nl.wikipedia.org/wiki/Eigentijd):

tau = t*sqrt (1 - v**2/c**2)

The two equations of the Lorentz Transformation are as follows (see Wikipedia):

x '= x / sqrt (1 - v**2/c**2) - v*t / sqrt (1 - v**2/c**2) (1)

and

t '= - ((v*x) / c**2) / sqrt (1 - v**2/c**2)) + t / sqrt (1 - v**2/c**2) (2)

Now x 'and t' represent points. x' represents a location (point in space) and t' is a time (point in time). One may write the two equations of the Lorentz Transformation for intervals as follows:

Delta x ' = Delta x / sqrt (1 - v**2/c**2) - v * Delta t / sqrt (1 - v**2/c**2) (3)

and

Delta t '= - ((v * Delta x) / c**2) / sqrt (1 - v**2/c**2)) + Delta t / sqrt (1 - v**2/c**2) (4)

We derive the formula for proper time by assuming, that Delta x '= 0. This is the case when Delta x = v * Delta t. Substituting this into formula (4) we obtain:

Delta t '= Delta t * sqrt (1 - v**2/c**2)

or

tau = t * sqrt (1 - v**2/c**2)

Note, that tau refers to a time interval.

---------------------------------------

Completely analogous to the derivation of the formula for the proper time, I derived the formula for the proper space (proper length). The formula is as follows (Wikipedia: http://en.wikipedia.org/wiki/Proper_length):

sigma = x * sqrt (1 - v**2/c**2) .

We derive the formula for proper space by assuming, that Delta t '= 0. This is the case when Delta t = (v * Delta x) / c**2. Substituting this into formula (3) we obtain:

Delta x '= Delta x * sqrt (1 - v**2/c**2)

or

sigma = x * sqrt (1 - v**2/c**2)

Note, that sigma refers to a space interval (one dimensional).

---------------------------------------

My question is: Are these derivations correct?
 
Last edited by a moderator:
Physics news on Phys.org
Is there anybody who might want to answer my question?
 
I noticed that you kept the meaning of sigma and tau deliberately vague. As in sigma is a space interval and tau is a time interval. You need to arrive at a result where proper length and coordinate length and proper time and coordinate time are related to each other. For example your conclusion for proper length is: a space interval is equal to a different space interval times some factor, but there is no mention of proper length anywhere. Now my question is what is the proper length in that result?
 
Last edited:
Cyosis said:
I noticed that you kept the meaning of sigma and tau deliberately vague. As in sigma is a space interval and tau is a time interval. You need to arrive at a result where proper length and coordinate length and proper time and coordinate time are related to each other. For example your conclusion for proper length is: a space interval is equal to a different space interval times some factor, but there is no mention of proper length anywhere. Now my question is what is the proper length in that result?

Proper length:

Delta x' = Delta x * sqrt (1 - v**2/c**2)

or

L' = L * sqrt (1 - v**2/c**2)

with L = Delta x and L' = Delta x'.

Proper space (along the x-axis only):

sigma = x' = x * sqrt (1 - v**2/c**2) .

Note, that sigma, x and x' can be negative, whereas L and L' cannot.

I hope this answers your question.
 
It does I think. You're saying that sigma is the proper length as I suspected. When you look at your formula it says that sigma is always smaller than x. Which is wrong, because the proper length is the longest length of the object.

I am short on time I will elaborate further later.
 
Last edited:
Cyosis said:
It does I think. You're saying that sigma is the proper length. When you look at your formula it says that sigma is always smaller than x. Which is wrong, because the proper length is the longest length of the object.

I am short on time I will elaborate further later.

A moving ruler (measuring stick) is observed shorter in the direction of the movement!

Might this be of any help?
 
A moving ruler (measuring stick) is observed shorter in the direction of the movement!

Might this be of any help?

Yes this is correct. However your derived equation, given that sigma is the proper length, says the opposite.
 
We are doing relativity here which makes it very important that you are very clear on what your formulae and notations mean. You also need to be clear on the different type of reference frames. Your derivation exists of writing down the Lorentz transformations then doing some algebraic manipulations. But you never really say what you're doing.

We derive the formula for proper time by assuming, that Delta x '= 0.

First of all this is not an assumption. Proper time has a very clear definition. Proper time is the time between two events in the inertial frame where these events happen at the same location.

Let us investigate an inertial frame S with coordinates (x,t) and an inertial frame S' with coordinates (x',t'), which are moving relative to each other with a speed v. If we measure an event at the coordinates (x_1',t_1') and a second event at (x_1', t_2') in frame S' then the proper time is \Delta t'=t_2'-t_1', because the two events happened at the same position in frame S'. Due to relativity of simultaneity these two events will happen at x_1 and x_2, with x_1 \neq x_2 in frame S.

Now if we measure the time between an event with coordinates (x_1,t_1) and an event with coordinates (x_1,t_2) in frame S then \Delta t =t_2-t_1 would be the proper time. As it should be otherwise there would be some absolute inertial frame which goes directly against relativity.

We derive the formula for proper space by assuming, that Delta t '= 0.

Again there is no assumption in the definition of proper length. The idea is similar to that of proper time.

Consider two inertial frames again, namely S and S', which are again moving at a speed v relative to each other. Now let us place a rigid object, with length \Delta x' in S' along the x'-axis, which is at rest in S'. Here \Delta x' is the proper length. We want to find its length in frame S. To measure its length we will need to mark its end points simultaneously in S. We call these markings events and in frame S these events happen at the same time. Therefore we can set \Delta t=0. If you carry out the derivation correctly you will find that \Delta x'>\Delta x.

What you have done is mix frames. You mark the endpoints simultaneously in frame S' and therefore set \Delta t'=0. So far so good. You then calculate \Delta x' and call this the proper length. This is wrong because S is the object's rest frame.
 
Last edited:
Cyosis said:
Let us investigate an inertial frame S with coordinates (x,t) and an inertial frame S' with coordinates (x',t'), which are moving relative to each other with a speed v. If we measure an event at the coordinates (x_1',t_1') and a second event at (x_1', t_2') in frame S' then the proper time is \Delta t'=t_2'-t_1', because the two events happened at the same position in frame S'.

I suppose t_1' < t_2'?
 
  • #10
Cyosis said:
Let us investigate an inertial frame S with coordinates (x,t) and an inertial frame S' with coordinates (x',t'), which are moving relative to each other with a speed v.

I suppose S' is moving with velocity v with respect to S or measured from S?
 
  • #11
Cyosis said:
Now if we measure the time between an event with coordinates (x_1,t_1) and an event with coordinates (x_1,t_2) in frame S then \Delta t =t_2-t_1 would be the proper time.

I suppose t_1 < t_2?
 
  • #12
Sure, although it doesn't really matter as long as one precedes the other in all inertial frames.
 
  • #13
To Cyosis

Your exposition is very clear. Thank you.
 
  • #14
Cyosis said:
If we measure an event at the coordinates (x_1',t_1') and a second event at (x_1', t_2') in frame S' then the proper time is \Delta t'=t_2'-t_1', because the two events happened at the same position in frame S'.

It looks to me, that this definition of proper time implies, that \Delta x' = 0. This is exactly what I did in deriving

Delta t '= Delta t * sqrt (1 - v**2/c**2)
 
Last edited:
  • #15
Yes your conclusion about proper time was correct, but not about proper length.
 
  • #16
Cyosis said:
Proper time has a very clear definition. Proper time is the time between two events in the inertial frame where these events happen at the same location.

The proper time between two events depends on the path taken between these events. Your definition is OK for an inertial path. All ideal clocks, whatever their motion, measure proper along their own worldlines, and its value is frame invariant, unlike coordinate time. In the inertial frame in which a clock is at rest, its worldline coincides with the time axis of that frame and so in this case, your scenario, proper time is equal to coordinate time.

Matheinste.
 
  • #18
kev said:
Hi Adven. This thread here https://www.physicsforums.com/showthread.php?t=393523 might be helpful to you. It is on a very similar topic.

Hi Kev,

I looked into it and it looks very interesting. I am going to study it in more detail. They seemed to follow the same approach as I did.

Thanks a lot for your information.
 

Similar threads

Undergrad The Lorentz factor gamma and proper time
  • · Replies 31 ·
2
Replies
31
Views
3K
Undergrad Are equations for spacetime intervals correct?
  • · Replies 6 ·
Replies
6
Views
1K
Undergrad Need to resort to spherical wavefront to derive the LTs?
  • · Replies 123 ·
5
Replies
123
Views
8K
Graduate Spacetime interval in Galilean relativity
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Are the Lorentz Transformations False?
  • · Replies 54 ·
2
Replies
54
Views
4K
High School Am I understanding the concept of proper frame of reference?
  • · Replies 33 ·
2
Replies
33
Views
2K
Undergrad What Is the Proper Time in Curved Space?
  • · Replies 48 ·
2
Replies
48
Views
6K
Undergrad A Hidden Zero in Einstein's Simple Derivation
  • · Replies 22 ·
Replies
22
Views
3K
Undergrad Where is the mistake in this derivation of Length Contraction?
  • · Replies 9 ·
Replies
9
Views
1K
Undergrad Understanding Lorentz Factor & Proper Time Invariance
  • · Replies 32 ·
2
Replies
32
Views
4K