Understanding Proper Time in Einstein's Special Theory of Relativity

[hschia]

Howdy! I'm new to Physics Forum and glad to be part of the community.

I just finished reading Einstein's Special Theory of Relativity and cannot understand the term 'proper time'. What does it specifically mean? When deriving the equation for length contraction, the author of Giancoli Sixth Edition mentioned the proper time as 'the time of departure of an astronaut to the time of arrival on planet Venus.' Why can't the proper time be the time of measured by an observer on Earth?

Thanks for your help!

 

[tiny-tim]

welcome to pf!

howdy, hschia! welcome to pf!

the proper time of an observer is the time shown on the observer's own clock

("proper" is another word for "own", as in "property")

… Why can't the proper time be the time of measured by an observer on Earth?

because that's someone else's time, not the astronaut's own time

 

[hschia]

Thanks for the explanation!

 

[Fredrik]

I prefer to define "proper time" mathematically, as a property of a timelike curve in spacetime, and then take one of the axioms that define SR to be "A clock measures the proper time of the curve in spacetime that represents its motion".

The simplest version of the mathematical definition is to say that if x:[a,b]→ℝ⁴ is a curve that represents an inertial observer's description of a particle's motion, i.e. a function such that x(t) is the particle's position coordinates in an inertial coordinate system S at time coordinate t in S, then the proper time of that curve is

$$\tau(x)=\int_a^b \sqrt{-g(x'(t),x'(t))}dt$$

where g:ℝ⁴→ℝ⁴ is the function that (if I use the convention to write vectors with 4 components in the form x=(x₀,x₁,x₂,x₃)) is defined by

$$g(x,y)=-x_0y_0+x_3y_3+x_3y_3+x_3y_3$$

(Note that there exist other conventions for where the minus signs should be, and where to put the indices (upstairs or downstairs)).

 

[Naty1]

Why can't the proper time be the time of measured by an observer on Earth?

Different observers moving at different relative velocities measure time differently. It's RELATIVE.

There is a good general description here explaining clock time AT events, called proper time, and clocktime of distant observers, call coordinate time.

http://en.wikipedia.org/wiki/Proper_time

One is neither better nor worse nor more accurate than the other,but they are in general DIFFERENT. This is part of Einstein's great insight that time is not constant as everyone had thought before special relativity (different clocks run at different speeds); instead, contrary to what everyone previously thought, both space and time, called "spacetime" as a result of Einsteins insights, are now known to be variable...THEY are not constant.

 

[G01]

Different observers moving at different relative velocities measure time differently. It's RELATIVE.

There is a good general description here explaining clock time AT events, called proper time, and clocktime of distant observers, call coordinate time.

http://en.wikipedia.org/wiki/Proper_time

One is neither better nor worse nor more accurate than the other,but they are in general DIFFERENT. This is part of Einstein's great insight that time is not constant as everyone had thought before special relativity (different clocks run at different speeds); instead, contrary to what everyone previously thought, both space and time, called "spacetime" as a result of Einsteins insights, are now known to be variable...THEY are not constant.

I like this way of thinking as well. Given two events, the proper time interval between the two is the time interval measured by the clock present at both events.

 

[Fredrik]

I like this way of thinking as well. Given two events, the proper time interval between the two is the time interval measured by the clock present at both events.

Did you mean "the longest possible time interval that a clock present at both events can have measured"? If not, it's wrong. Proper time is a property of a curve, so it's not determined by a pair of events unless you say e.g. that you're talking about the curve that maximizes the proper time.

 

[G01]

Did you mean "the longest possible time interval that a clock present at both events can have measured"? If not, it's wrong. Proper time is a property of a curve, so it's not determined by a pair of events unless you say e.g. that you're talking about the curve that maximizes the proper time.

Ah, yes. Good call.

Then how about: "The proper time for a given observer is measured by the clock that travels with the observer."

Then the clock is always at rest relative to the observer in question and thus travels the same worldline.

Does this statement work, or am I still missing something?

 

[Fredrik]

That sounds good. What you might want to think about is just if you want your statement to be the definition of the term "proper time" or the axiom of SR that tells us how to measure proper time (which would then be defined by the integral in #4 or its generalization to other coordinate systems and other spacetimes).

 

[G01]

That sounds good. What you might want to think about is just if you want your statement to be the definition of the term "proper time" or the axiom of SR that tells us how to measure proper time (which would then be defined by the integral in #4 or its generalization to other coordinate systems and other spacetimes).

Good idea. What I am going for in not the definition of proper time, but, as you say, an axiom that can be used by new SR students to measure proper time with less confusion, but is still consistent with the rigorous definition.

 

[Naty1]

G01, Fredrik: much better than the WIKI description:

"The proper time for a given observer is measured by the clock that travels with the observer."

Then the clock is always at rest relative to the observer in question and thus travels the same worldline.

One of you guys should submit that change to Wiki!...there are a lot of such improvements that can be made there...

 

[Naty1]

I saved this definition from somewhere here in the forums:

The proper time of an object is the time recorded by a clock carried by the object.

I'm thinking that if an observer and and object travel between two points at different speeds ...or take different paths to those observation points...proper time for the object and observer will differ...

So is one explanation better than the other or do we need both...??

 

[Fredrik]

The exact statement that I eventually settled on is the one in italics in the first paragraph of #4. I think the way I'm saying it makes it clear that it's not a definition of the term, but an axiom of special (and general) relativity. (The rest of that post is about how to define the term).

The statement quoted in #12 sounds like a definition of proper time that associates that term with something in the real world instead of something mathematical. Most people seem to prefer to define proper time that way. I don't, because if we define it that way, then we need another statement to make the connection between the mathematics and the real world.

 

[hschia]

I prefer to define "proper time" mathematically, as a property of a timelike curve in spacetime, and then take one of the axioms that define SR to be "A clock measures the proper time of the curve in spacetime that represents its motion".

The simplest version of the mathematical definition is to say that if x:[a,b]→ℝ⁴ is a curve that represents an inertial observer's description of a particle's motion, i.e. a function such that x(t) is the particle's position coordinates in an inertial coordinate system S at time coordinate t in S, then the proper time of that curve is

$$\tau(x)=\int_a^b \sqrt{-g(x'(t),x'(t))}dt$$

where g:ℝ⁴→ℝ⁴ is the function that (if I use the convention to write vectors with 4 components in the form x=(x₀,x₁,x₂,x₃)) is defined by

$$g(x,y)=-x_0y_0+x_3y_3+x_3y_3+x_3y_3$$

(Note that there exist other conventions for where the minus signs should be, and where to put the indices (upstairs or downstairs)).

What does this mathematical equation of proper time mean?

 

[Fredrik]

What does this mathematical equation of proper time mean?

It would help if you said something about what mathematics you know. Do you know what an integral is?

It means roughly that you need to break up the curve in short segments, find the difference dt between the time coordinates (in an inertial coordinate system) of the endpoints of the segments, find the corresponding differences dx, dy, dz for the position coordinates (same coordinate system), calculate

$$\sqrt{dt^2-dx^2-dy^2-dz^2}=dt\sqrt{1-\vec v\,^2}=\frac{dt}{\gamma}$$

for each segment, and add up the results.