Proper Time

[DrRocket]

This was intended as a post in the thread bon spacetime in special relativity. However the forum software (or me) resulted in the piece being superimposed over the initial post, much editing and extreme confusion. So here it is in a separate thread. It is self-contained and therefore a bit redundant to the spacetime thread.

 

The purpose is to show why "proper time" defined in terms of the "spacetime interval" has anything to do with what clocks measure.

 

This explanation is based on a geometric treatment of special relativity. It is largely based on the treatment given in the book The Geometry of Minkowski Space time by Gregory L. Naber. You can refer to that book for more detailed mathematics and to Essential Relativity, Special General and Cosmological by Wolfgang Rindler for a less mathematical and more physical treatment. Some of Rindler's perspective is also a part of this note.

 

This argument carries over unchanged to general relativity simply by a localization observation -- the metric of GR is locally the Minkowski metric.

 

 

What is Minkowski spacetime ?

 

Minkowski spacetime is the setting for special relativity. It is by definition ordinary 4-space with a non-degenerate quadratic form of signature (+,-,-,-). Equivalently one can use a quadratic form of signature (-,+,+,+) and this is the convention used by Naber, but we will use the other convention here.

 

The quadratic form defines an inner product on Minlkowski space. It is analogous to the dot product of ordinary Euclidean space, except that it is not positive-definite. This means that It is possible for the inner product (squared length) of a vector with itself to be negative , and it is possible for the inner product of a vector with itself to be zero even if the vector is not the zero vector. So, in Minkowski space a nonn-zero vector can be perpendicular to itself. That fact requires you to readjust your intuition with regard to some geometric ideas, so don't get blindsided by some of this weirdness.

 

Now just as with the ordinary inner product, there is no a priori need to define a basis so as to express it as a "dot product". To take the ordinary product of two vectors in 2-space just form the product of their lengths and the cosine of the angle between them – no basis needed to do this geometrically. So think of the Minkowski inner product as a geometric idea, and we'll talk about the relationship to a basis set.

It takes a little more work than in the usual case of a positive-definite inner product, but one can show that given a non-degenerate inner product one can find an orthonormal basis for the underlying vector space. In this more general case an orthonormal basis is a basis in which distinct elements are orthogonal (have inner product 0) and in which the inner product of any basis element with itself is 1 or -1. One can prove and any two orthonormal basis sets always have the same number of elements with inner product with themselves equal to -1, and that defines the "signature" of the quadratic form. In the case of Minkowski space the signature is (+,-,-,-). The inner product of a vector with itself is called the "squared norm". A vector with a negative squared norm is called "space-like" and one with a positive squared norm is called "time-like".

 

We will denote the inner product, using the Minkowski quadratic form, of 4-vectors X and Y by <X,Y> and then then length of a vector X is the norm of X,

 

[math] |X|=\sqrt{<X,X>}[/math]

 

Transformations that preserve the inner product are (inhomogenous) Lorentz transforms, sometimes called Poincare transforms. One generally restricts attention to a subset of the full set of Lorentz transforms for physical reasons, but that is a subject for another time. Lorentz transforms correspond to individual observers and serve to relate coordinate measurement for one observer to another observer. Objects that are preserved by Lorrentz transforms are called invariants of special relativity.

 

For the purposes herein we will work in units in which the speed of light is 1. That makes the usual formula for gamma simply

[math] \gamma=\frac {1}{\sqrt {1-v^2}}[/math]

 

Length in Minkowski space The length of a vector X is just |X|. The length of a curve is given in the usual way. A parameterized curve in Minkowski space is just a function from the real line, or a line segment taking as its values 4-vectors in Minkowski space. Let [math] \phi [/math] be such as curve, defined on [0,1]. Then the length of [math] \phi [/math] (arc length)is just

 

 

[math] \int_0^1 | \frac {d \phi (\tau}{d \tau}| d \tau[/math]

 

as in the case of ordinary Eudlidean space with a positive definite inner product, one can define an arc length parameter for φ, call it s by

 

(http://www.math.hmc....es_and_arc.html)

 

 

[math] s(t) =\int_0^t |\frac {d \phi}{d \tau}| d \tau[/math]

 

 

One can parameterize a curve using arc length, and one finds then that the "speed" along the curve is simply 1.

 

 

Proper Time in Minkowski Space

 

The proper time separating the end points of a curve in Minkowski space is simply the length of the curve, and the proper time parameter, τ, is simply arc length. This is a definition.

The obvious question is what the definition of "proper time" , τ , has to do with "time", t, since t Is what is measured by the clock of an observer and τ on the surface is nothing but distance associated with an unconventional notion of "length". So far we have worked purely in terms of mathematics and the geometry of Minkowski space. To address this new question requires physical reasoning. \

Consider a curve in Minkowski space that consists of short displacements in space at constant speed. Any smooth curve is approximable by such a curve. This curve represents the trajectory of a particle in Minkowski space, and in the reference frame of that particle we select an orthonormal basis x,y,z,t.

Now consider one increment of displacement, from[math] (x_0, y_0,z_0,t_0 )=X_0[/math] to [math](x_1,y_1,z_1,t_1 )=X_1[/math] where the displacement is timelike the length of the displacement τ is just

 

 

[math] T=|X_1-X_0 |=\sqrt {t_1-t_0)^2-(x_1-x_0)^2-(y_1-y_0 )^2-(z_1-z_0)^2}[/math]

 

 

=[math]\sqrt{\frac{1-((x_1-x_0)^2+(y_1-y_0 )^2+(z_1-z_0)^2)}{(t_1-t_0)^2 }} (t_1-t_0) [/math]

 

=[math] \sqrt {1-v^2 } \Delta t [/math]

 

=[math] \frac {1}{\gamma} \Delta t[/math]

 

Or

 

[math]\Delta t=\gamma T[/math]

 

This shows that τ is the time sensed by a clock that is co-moving with the particle on this small segment. Since any smooth curve is approximated by a series of such segments it follows that the arc length along the curve is identifiable with the time experienced by a particle moving along the cure. So the parameter τ is deserving of the term "proper time". Note that proper time is defined geometrically, and since it is preserved by Lorentz transforms, it is an invariant of the theory

[michel123456]

I never use the watch topic feature. I dont want notifications, only to be kept on a list, about like a favorite in Internet Explorer.

(this has nothing to do with this thread, see moderator note below (edited by michel)

 

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

 

Since DrRocket seems to know a lot, he may be interested in an older thread of mine.

It is about time "sensed by a clock that is co-moving", as we do when standing at rest.

 

The thread was "Rod at rest" with an interesting (to me) dialogue with Swansont who disagreed.

 

http://www.scienceforums.net/topic/53547-rod-at-rest/page__p__578834__fromsearch__1#entry578834

Edited February 20, 2011 by michel123456

[ajb]

You can refer to that book for more detailed mathematics and to Essential Relativity, Special General and Cosmological by Wolfgang Rindler for a less mathematical and more physical treatment.

 

I just can't really get on with Rindler's book. It is just not mathematical enough for me to get to grips with what he is trying to tell us.

[DrRocket]

I just can't really get on with Rindler's book. It is just not mathematical enough for me to get to grips with what he is trying to tell us.

 

It is a matter of taste, personal perspective and inclination. I tend toward your viewpoint and personally much prefer Naber's treatment. But it sometimes helps to see the perspective of "the other guy" as well.

 

Rindler can turn on the mathematics when he wants to, and is a far more important figure in relativity than is Naber. Rindler's two volumes, with Penrose, Spinors ans space-time ought to be mathematical enough for either of us.

[swansont]

!

Moderator Note

Discussion on board notification/watching details moved to support http://www.scienceforums.net/topic/55037-watching-topics/

[michel123456]

Good move.

[md65536]

Is proper time equivalent to "local time in the frame of reference of an observer, for events that happen at the location of the observer"?

 

And local time (is there a better term?) is a bit more general, because it can also be used to describe the timing of remote events?

 

And is coordinate time then basically time according to any clock that is remote from the observer?

 

 

Is it correct to use these phrases when speaking of relativistic scenarios? Eg. a traveling observer's ideal clock measures proper time and always ticks at a constant rate. For other observers the same clock measures coordinate time, which ticks at a variable rate in general.

 

Are there other related or better terms for describing time according to various observers and clocks?

[IM Egdall]

See link for explanation of proper time and coordinate time:

 

http://en.wikipedia....Coordinate_time

 

I think you are correct on proper time. It is, in fact, the spacetime interval between two events in units of time. (The square of the spacetime interval equals the difference between the square of the time interval and the square of the space interval). And the spacetime interval is absolute, in that it is unaffected by relative uniform motion.

 

Consider two events that happen at different times and at different locations in space. Say you are present at both events. For example

Event 1: you leave the Earth in your rocket.

Event 2: You arrive on Mars.

 

From your point-of-view (reference frame), assuming uniform motion, you are at rest and Mars comes to you. So in your reference frame, the space interval you measure is zero!

 

Thus since there is no space interval to subtract, the spacetime interval is simply equal to the time interval between the two events. This for you is the time on your wristwatch from event 1 to event 2. So proper time is also called wristwatch time.

 

Co-ordinate time is what someone who stays on Earth would measure for the time interval between your leaving Earth (event 1) and arriving on Mars (event 2). This observer is not present at both events (only at event 1). Co-ordinate time is not absolute. It is affected by relative uniform motion.

Edited August 6, 2011 by I ME

[md65536]

Co-ordinate time is what someone who stays on Earth would measure for the time interval between your leaving Earth (event 1) and arriving on Mars (event 2). This observer is not present at both events (only at event 1). Co-ordinate time is not absolute. It is affected by relative uniform motion.

The observer on Earth would use my clock (which travels to Mars) to measure coordinate time, correct?

 

If they use their own clock, is that wristwatch time? It couldn't count as proper time. Or would it be coordinate time also? I was calling this "local time" but that seems a misleading phrase for measuring the time of remote events (using a local wristwatch).

 

We could say that to an Earthbound observer, the time between my leaving and the observation on Earth of my arrival on Mars, can be measured in proper time.

 

 

 

[DrRocket]

Is proper time equivalent to "local time in the frame of reference of an observer, for events that happen at the location of the observer"?

 

And local time (is there a better term?) is a bit more general, because it can also be used to describe the timing of remote events?

 

And is coordinate time then basically time according to any clock that is remote from the observer?

 

 

Is it correct to use these phrases when speaking of relativistic scenarios? Eg. a traveling observer's ideal clock measures proper time and always ticks at a constant rate. For other observers the same clock measures coordinate time, which ticks at a variable rate in general.

 

Are there other related or better terms for describing time according to various observers and clocks?

 

Clocks measure the proper time of their world line. That is all that any clock measures.

 

Coordinate time is a fiction. It is time as modeled in some choice of a coordinate patch, and would correspond to the time registered by a real clock only in the case of flat spacetime. It is an approximation that is reasonably accurate in approximately flat circumstances; i.e. away from large gravitational fields over moderate distances. Only coordinate time makes sense as a comparison of separated clocks. Special relativity, being a theory that excludes gravity, deals with a situation in which coordinate time and proper time coincide, but in the real world the time of special relativity is coordinate time -- SR is the local approximation to GR.

[md65536]

Clocks measure the proper time of their world line. That is all that any clock measures.

 

Coordinate time is a fiction. It is time as modeled in some choice of a coordinate patch, and would correspond to the time registered by a real clock only in the case of flat spacetime. It is an approximation that is reasonably accurate in approximately flat circumstances; i.e. away from large gravitational fields over moderate distances. Only coordinate time makes sense as a comparison of separated clocks. Special relativity, being a theory that excludes gravity, deals with a situation in which coordinate time and proper time coincide, but in the real world the time of special relativity is coordinate time -- SR is the local approximation to GR.

Isn't the proper time of one observer a coordinate time of another observer?

 

I don't understand how curved spacetime prohibits correspondence to a real clock.

Any observations of a distant clock (or signals from one) would arrive via a geodesic, which would define a distance to the clock, and thus a specific time delay of the observation, regardless of curvature. So it seems that the observer could calculate the time registered by the real clock -- and wouldn't this correspond to the coordinate time?

 

EXCEPT... the time delay would be measured in proper time. With SR you could just divide by gamma to find the delay in terms of coordinate time? Is there no similar thing in GR?

 

 

[DrRocket]

Isn't the proper time of one observer a coordinate time of another observer?

 

no

 

I don't understand how curved spacetime prohibits correspondence to a real clock.

Any observations of a distant clock (or signals from one) would arrive via a geodesic, which would define a distance to the clock, and thus a specific time delay of the observation, regardless of curvature. So it seems that the observer could calculate the time registered by the real clock -- and wouldn't this correspond to the coordinate time?

 

This makes no sense.

 

Geodesics in general relativity are determined by the Lorentzian metric. It is a metric on spacetime, not just space, and in units for which c=1, the length os a timelike geodesic is in fact the proper time of that segment of a world line. You measure "distance" withy a clock.

 

 

EXCEPT... the time delay would be measured in proper time. With SR you could just divide by gamma to find the delay in terms of coordinate time? Is there no similar thing in GR?

 

SR is GR in flat Minkowski spacetime. There, coordinate time and proper time are the same thing. This is because the spacetime manifold and its tangent space at any point are isometric.

[md65536]

SR is GR in flat Minkowski spacetime. There, coordinate time and proper time are the same thing. This is because the spacetime manifold and its tangent space at any point are isometric.

I don't think we're talking about the same thing here. Coordinate time and proper time of events are the same thing in SR only according to clocks that are at the same location of the events (or relatively at rest and synchronized to the observer's clocks) [second paragraph of http://en.wikipedia....Coordinate_time].

 

When we speak of time dilation, we're speaking of one clock ticking at a different rate relative to another clock.

http://en.wikipedia..../Lorentz_factor

The former refers to coordinate time (is this incorrect?), the latter to proper time.

 

 

If I'm using the term coordinate time incorrectly, then what term should be used instead to describe the time according to a remote moving clock that ticks slower relative to "wristwatch time"?

 

 

 

 

Edit: I think I see my mistake...s...

- Coordinate time and proper time don't inherently refer to different clocks. In SR, the coordinate time and proper time of a given clock and observer are the same.

- Each clock will have its own proper time.

- The time of a remote moving clock is just the "time of that clock according to the observer"... it doesn't have a special name.

Edited August 7, 2011 by md65536

[DrRocket]

 

Edit: I think I see my mistake...s...

- Coordinate time and proper time don't inherently refer to different clocks. In SR, the coordinate time and proper time of a given clock and observer are the same.

- Each clock will have its own proper time.

- The time of a remote moving clock is just the "time of that clock according to the observer"... it doesn't have a special name.

 

Now you are getting it.

 

Even in special relativity what one observer views as the time coordinate of spacetime will differ from that of an observer in relative motion, so "coordinate time" varies with the observer -- different observers coordinatize spacetime differently. But the coordinate time of an observer will be the proper time of that observer (or clock) -- you are just talking about a spacetime interval with a constant sparial coordinate in the frame of the observer.

 

The difference between SR and GR is that in SR there is a single global set of coordinates. In GR, with a curved spacetime, coordinates are only local.

 

The key to all of this is that proper time is associated with a timelike curve (aka world line) in spacetime and is just the length of that curve, divided by c, measured in the spacetime metric. A consequence of this is that the 4-velocity of anything, not just light, is c (velocity is distance divided by time, and the distance along a curve is just the time divided by c).

 

Time in relativity is a different animal from time in Newtonian mechanics. Time and space are not separate, but rather are intertwined in spacetime. They are only local concepts in general relativity, and are observer dependent in both SR and GR. There is no absolute time and no absolute space.

 

This thread on spacetime in special relativity might be helpful.

http://www.scienceforums.net/topic/54988-spacetime-in-special-relativity/

[time]

!

Moderator Note

Post deleted; it is basically the same as http://www.scienceforums.net/topic/58996-time/

time, promoting your own nonstandard views is hijacking and against the rules. Discuss your topic in the linked thread, and only in the linked thread. This discussion is about proper time in the context of the theory of relativity