A question on proper time in special relativity

[user1139]

Homework Statement:: This isn't a homework but more of a conceptual question.
Relevant Equations:: Proper time, $\tau$

Simply put, the proper time between two events as observed in an unprimed frame is calculated along the timelike worldline between the two events. This implies that the spatial primed spatial coordinates are zero which further implies that only the spatial primed time coordinates remain.

With the above, here are my questions:

On the spacetime diagram, does it mean that the two events will then always lie on the primed time axis? What if in the unprimed frame, the two events are situated such that the same two events do not lie on the primed time axis only? How should the proper time be calculated then?

 

[Ibix]

Simply put, the proper time between two events as observed in an unprimed frame is calculated along the timelike worldline between the two events.

Corrected that for you. Proper time is analogous to the length of a worldline, and the length is the length whatever frame you use.

This implies that the spatial primed spatial coordinates are zero which further implies that only the spatial primed time coordinates remain.

This doesn't make sense, even dropping the two extraneous "spatials". It is true that for any two timelike separated events there exists a frame in which their spatial separation is zero and the coordinate time between them is equal to the proper time between them. This doesn't mean that only the time coordinates remain.

On the spacetime diagram, does it mean that the two events will then always lie on the primed time axis? What if in the unprimed frame, the two events are situated such that the same two events do not lie on the primed time axis only?

If the two events are timelike separated there exists a frame in which they are in the same place, but this doesn't mean that they must lie on the time axis, if that's what you are asking. And they are in different places in every frame except one, anyway.

How should the proper time be calculated then?

Do you know the formula for calculating the proper time between two events?

 

[user1139]

Yes, I know the formula for calculating proper time but I am confused as to what it means to set the primed spatial coordinates to zero when finding the proper time.

The attached photo is what I am referring to when I mention primed and unprimed coordinates.

 

[Ibix]

They aren't setting the coordinates to zero. They're observing that if the events are at the same place then the difference between their spatial coordinates is zero. $x'$ can be whatever it wants, as long as it's the same for both so that $\mathrm{d}x'=0$.

 

[user1139]

Oh so does that mean that on the spacetime diagram the two events will be on the same line that is parallel to the primed time coordinates but it need not be the case that $x'=0$?

 

[Ibix]

Yes.

Lorentz transforms are closely analogous to rotations. If you rotate a line so that it lies parallel to the $y$ axis then $\Delta y$ between the ends of the line is equal to its length, whether it's on the axis or not. The same applies to worldlines between events - if you Lorentz transform them so that they lie parallel to the $t'$ axis then $\Delta t'$ between the ends of the worldline is equal to its proper time, whether it's on the axis or not.

 

[user1139]

I see. Does the spacetime diagram scenario we talked about also imply that it is always possible to set up the prime frame such that the two events lie on a line parallel to $t'$? And if so then the usual primed axes we see need not always be the case?

By usual primed axes I mean the following:

 

[PeroK]

Yes, I know the formula for calculating proper time but I am confused as to what it means to set the primed spatial coordinates to zero when finding the proper time.

The attached photo is what I am referring to when I mention primed and unprimed coordinates.

View attachment 284706

That's defining proper time as coordinate time in the primed frame where the object is at rest. And, owing to the invariance of the spacetime interval, proper time is seen to be the length of the spacetime interval in any, arbitrary frame. That's the first line.

The rest is some mathematics to show that it is the integral of $\frac 1 \gamma$ (in any frame).

Note that everything here holds where $v$ is a function of $t$. And we have:
$$\Delta \tau = \int_c \frac 1 {\gamma(t)} dt$$
So, in your unprimed $t-x$ spacetime diagram, the object can have any timelike motion.

 

[Ibix]

I see. Does the spacetime diagram scenario we talked about also imply that it is always possible to set up the prime frame such that the two events lie on a line parallel to $t'$?

If the two events are timelike separated, yes. If they are null or spacelike separated, no.

And if so then the usual primed axes we see need not always be the case?

The green axes aren't necessarily at that angle, if that's what you mean. The $ct'$ axis will always be timelike, which means that on a Minkowski diagram it will always lie less than (never equal to) $\pm 45^\circ$ from the $ct$ axis. The angle it makes with the time axis will always be the same as the angle the $x'$ axis makes with the $x$ axis.

 

[user1139]

Is it then correct to say that:

a) Two timelike separated events are found within the lightcone and the worldline connecting the two events is a timelike worldline. The timelike worldline is where all of the spacetime points on it can be arranged to represent a sequence of events all taking place at the same place but at different times. This will then imply that timelike separated events are causally connected.

b) For two spacelike separated events, they are found outside the lightcone and the worldline connecting the two events is a spacelike worldline. The spacelike worldline is where all of the spacetime points on it can be arranged to represent a sequence of events all taking place at the same time but at different places. This will then imply that spacelike separated events are not causally connected.

 

[Ibix]

Two timelike separated events are found within the lightcone

Not quite. There is no "the" lightcone. There are two lightcones associated with every event in spacetime - a future lightcone and a past lightcone. For two timelike separated events the second is in the future lightcone of the first and the first is in the past lightcone of the second.

The timelike worldline is where all of the spacetime points on it can be arranged to represent a sequence of events all taking place at the same place but at different times.

Not exactly. The timelike worldline is the set of events on a straight line between the two events (warning: you can have curved worldlines, but for the time being we're only talking about straight lines). You can find a frame where all events on a straight worldline are all in the same place, yes, but I wouldn't phrase it the way you have. Your way kind of implies that you can move events, but a frame change isn't moving events (that isn't possible) but rather changing the axes.

This will then imply that timelike separated events are causally connected.

They can be causally connected, to be pedantic. The fact that I wrote this means that you can read it at an event timelike separated from the one I'm at. But you could have chosen not to read, in which case I haven't affected you although in principle I could have done.

For two spacelike separated events, they are found outside the lightcone and the worldline connecting the two events is a spacelike worldline. The spacelike worldline is where all of the spacetime points on it can be arranged to represent a sequence of events all taking place at the same time but at different places.

Ditto previous comments about lightcones and frame changes.

This will then imply that spacelike separated events are not causally connected.

Correct.

Note that there are also lightlike (a.k.a. null) separated events, which lie on, not in or outside, the future/past lightcone of the other.

 

[user1139]

After reading your comments, I tried to apply them to understanding an inertial observer and an observer with proper acceleration in Minkowski spacetime in order to study the Unruh Effect. However the more I try, the more confused I get.

I’m confused about the following:

1) How do I go about understanding the world lines and light cones of events for these two observer?

2) Why is it that the proper time of the accelerated observer can be related to the inertial observer?

 

[PeroK]

After reading your comments, I tried to apply them to understanding an inertial observer and an observer with proper acceleration in Minkowski spacetime in order to study the Unruh Effect. However the more I try, the more confused I get.

I’m confused about the following:

1) How do I go about understanding the world lines and light cones of events for these two observer?

2) Why is it that the proper time of the accelerated observer can be related to the inertial observer?

You should post this as a homework problem, perhaps. In any case, you need to post more precisely what it is that you cannot calculate.

 

[user1139]

I’m trying to understand the concept of world lines, timelike separated events and light cones as applied to an inertial and an accelerated observer in Minkowski spacetime.

This is because I don’t understand how is it possible that the proper time of the accelerated observer can be related to the coordinate time of an inertial observer when the two observers are spacelike separated.

 

[PeroK]

I’m trying to understand the concept of world lines, timelike separated events and light cones as applied to an inertial and an accelerated observer in Minkowski spacetime.

Right, but that's like saying "I'm trying to learn differential calculus". That's not a specific problem. Short of posting a recommended text-book on SR, what can I post here to help you with something as broad as that?

This is because I don’t understand how is it possible that the proper time of the accelerated observer can be related to the coordinate time of an inertial observer when the two observers are spacelike separated.

This makes no sense. Observers are not spacelike separated; events may be spacelike separated.

In Minkowski space, the entire universe is described in the coordinates of any inertial reference frame. In other words, inertial reference frames are global.

You clearly have some fundamental misunderstandings of these concepts. You need to go back, revise the concepts and try to iron out the misconceptions.

 

[Ibix]

This is because I don’t understand how is it possible that the proper time of the accelerated observer can be related to the coordinate time of an inertial observer when the two observers are spacelike separated.

You are mixing up frames and observers. Easy enough, since they are often conflated, but they are not the same thing.

An observer is at a place in space and follows some worldline through spacetime. They do not have a coordinate time, only their own proper time, the interval along their own worldline.

A frame is an infinite family of observers, one at every place in space, all following parallel worldlines, all carrying a three number spatial coordinate, and all agreeing a mechanism for when they should zero their watches. Then the proper time since zero of one of this family is this frame's coordinate time at their location.

This family of observers is usually only notional, but it answers your problem: there's always a member of the family at any event whose watch and location you can read. In practice, you don't have infinitely many observers. You have a few who are equipped with binoculars and radar and what have you and calculate the coordinates of an event after the fact based on their observations.

This last point crosses over with PeroK's point. Since observers are worldlines, they cannot be spacelike or timelike separated in flat spacetime. Events on their worldline can be spacelike separated, but an event on your worldline now is inside the past lightcone of an event in my future - which is a high falutin' way of saying I can't see what you are doing now, but I will be able to see it a short time later (assuming there is nothing blocking line of sight etcetera etcetera).

 

[Dale]

I’m trying to understand the concept of world lines, timelike separated events and light cones as applied to an inertial and an accelerated observer in Minkowski spacetime.

A worldline is simply a curve in spacetime. For example, an observer’s worldline is the curve representing all of the events at which the observer was (or is or will be) located.

Before going into timelike and light cones, it is probably better to get a solid understanding of the spacetime interval itself.

The spacetime interval is given by $ds^2=-c^2 d\tau^2=-c^2 dt^2+dx^2+dy^2+dz^2$. Hopefully it is at least plausible that this represents a sort of “distance” in spacetime. When $ds^2>0$ the interval is called “spacelike”, and when $d\tau^2>0$ the interval is called “timelike”, and if $ds^2=d\tau^2=0$ the interval is called “lightlike” or “null”.

For every event on every worldline we can construct a tangent vector. This is exactly analogous to the tangent vector for an ordinary curve in space. If that tangent vector is timelike at every event on a worldline then the worldline is timelike. An accelerated observer therefore has a timeline worldline.

 

[Pyter]

As an aside, I've always found the notations ds² and dτ² mighty confusing: you see them squared and you assume they're always >= 0, while they can actually be negative. It would be clearer to write them as <ds, ds> or something along these lines.

 

[Sagittarius A-Star]

As an aside, I've always found the notations ds² and dτ² mighty confusing: you see them squared and you assume they're always >= 0, while they can actually be negative. It would be clearer to write them as <ds, ds> or something along these lines.

The ds can become imaginary and you can call it only "proper time" dτ, as long it is real and not 0. I don't see a problem with it. Under this condition:

dτ² = -ds² for (-+++) convention
dτ² = +ds² for (+---) convention

 

[vanhees71]

I'd define "proper time" exclusively as a quantity parametrizing a time-like world line, which physically can describe the world line of a massive particle. Then all these confusions in this thread do not occur. It's called "proper time", because it's the time an (ideal) clock fastened to the particle shows. For a time-like world line by definition
$$\mathrm{d} \tau=\frac{1}{c} \sqrt{\eta_{\mu \nu} \mathrm{d} x^{\mu} \mathrm{d} x^{\nu}},$$
using the west-coast-metric convention, i.e., (+---). By definition $\mathrm{d} \tau$ is thus a real and positive quantity.

 

[Pyter]

The ds can become imaginary

Never heard of that. AFAIK the only advocate of imaginary numbers in SR was Poincaré, but they decided to drop them later on.

 

[ergospherical]

It's much ado about nothing because $ds^2$ is merely notation. In co-ordinate basis $g = g_{\mu \nu} dx^{\mu} \otimes dx^{\nu}$ which is commonly abbreviated to $ds^2 = g_{\mu \nu} dx^{\mu} dx^{\nu}$, i.e. omitting "$\otimes$" and using the notation $ds^2$ instead of $g$ to, as Wald puts it, convey the "intuitive flavour" of an infinitesimal squared distance being determined by the coordinate separations.

In other words it's not the square of a number, just another symbol for $g$ [i.e. $ds^2(u,v) = g(u,v)$].

 

[Ibix]

As ergospherical says, $ds^2$ shouldn't be taken too literally as "a thing squared" (Carroll agrees with Wald, if memory serves). Generally if you need to take the square root (to integrate and get proper time or spatial distance for example), take $\sqrt{|ds^2|}$.

 

[Dale]

I'd define "proper time" exclusively as a quantity parametrizing a time-like world line

I would also. I don’t know if there is an authoritative source for that approach, but that has always been my understanding and approach.

 

[Pyter]

$ds^2$ shouldn't be taken too literally as "a thing squared" ##.

I know, but every time I have to remember that even though it's the same symbol at the denominator of a second derivative, for instance, in the SR context has another meaning.

 

[vanhees71]

I would also. I don’t know if there is an authoritative source for that approach, but that has always been my understanding and approach.

We don't need any authority in physics. It simply only makes sense for time-like curves! I don't know any textbook or paper, where it's used with other meaning.

 

[robphy]

I'd define "proper time" exclusively as a quantity parametrizing a time-like world line

I would also. I don’t know if there is an authoritative source for that approach, but that has always been my understanding and approach.

We don't need any authority in physics. It simply only makes sense for time-like curves! I don't know any textbook or paper, where it's used with other meaning.

The authority might be one who came up with the term "proper time" (translated from the original)

Let us now imagine a worldpoint $P(x, y, z, t)$ through which the worldline of a substantial point is passing, then the magnitude of the timelike vector dx, dy, dz, dt along the line will be
$$d\tau = \frac{1}{c}\sqrt{ c^2\ dt^2 − dx^2 − dy^2 − dz^2}.$$
The integral$\int d\tau = \tau$ of this magnitude, taken along the worldline from any fixed starting point $P_0$ to the variable end point P, we call the proper time of the substantial point at P.

https://www.minkowskiinstitute.org/mip/MinkowskiFreemiumMIP2012.pdf#page=57
(on text page 47, pdf-page 57 from this translation)

 

[Sagittarius A-Star]

The authority might be one who came up with the term "proper time" (translated from the original)
https://www.minkowskiinstitute.org/mip/MinkowskiFreemiumMIP2012.pdf#page=57
(on text page 47, pdf-page 57 from this translation)

H. Minkowski also proposed to shift the "i" issue from the spacetime interval to the unit system by defining the ratio between the unit of distance and the unit of time to be imaginary:
$$3 \cdot 10^5 \ km = \sqrt{-1} \ seconds$$
Source: same link, page 50, PDF page 60:
https://www.minkowskiinstitute.org/mip/MinkowskiFreemiumMIP2012.pdf#page=60

But this proposal is not accepted today. Maybe, because it would only shift the "i" issue from the spacetime interval (were it does not harm) to everywere else in physics.

 

[robphy]

H. Minkowski also proposed to shift the "i" issue from the spacetime interval to the unit system by defining the ratio between the unit of distance and the unit of time to be imaginary:
$$3 \cdot 10^5 \ km = \sqrt{-1} \ seconds$$
Source: same link, page 50, PDF page 60:
https://www.minkowskiinstitute.org/mip/MinkowskiFreemiumMIP2012.pdf#page=60

But this proposal is not accepted today. Maybe, because it would only shift the "i" issue from the spacetime interval (were it does not harm) to everywere else in physics.

True, but like lots of things in physics, some ideas persist and some don't.
The shift to "i" is a convention.

Ok, Here's something a little more recent

7.1 Let $\gamma$ be a causal trip. Define the length (i.e., "proper time")
of $\gamma$ to be:
$$\ell(\gamma)=\sum_{i=1}^k \{ \Phi(p_{i-1},p_i) \}^{1/2}$$
where successive segments of $\gamma$ are $p_0p_1$, $p_1p_2$, $\cdots$$p_{k-1}p_k$ (each segment $p_{i-1}p_i$
for definiteness, lying within a simple region $N_i$) and where $\Phi$ is the world function
defined in 2.13. (We have $\Phi(p_{i-1},p_i)\geq 0$ since $p_{i-1},p_i$ is causal.) This definition
simply assigns the obvious meaning of length, according to the space-time metric,
to any causal trip. Clearly $\ell(\gamma) > 0$ unless $\gamma$ consists entirely of null segments.

2.13. Definition. Let N be a simple region and define
the world-function $\Phi: N\times N \rightarrow R$ by $\Phi(x,y)=g(\exp_x^{-1}(y),\exp_x^{-1}(y))$; in other words,
$\Phi(x,y)$ is the squared length of the geodesic $xy$. Clearly $\Phi(x,y)=\Phi(y,x)$ and is
positive, negative or zero according as $xy$ is timelike, spacelike or null.

Maybe I can find one from 2021.

 

[dextercioby]

H. Minkowski also proposed to shift the "i" issue from the spacetime interval to the unit system by defining the ratio between the unit of distance and the unit of time to be imaginary:
$$3 \cdot 10^5 \ km = \sqrt{-1} \ seconds$$
Source: same link, page 50, PDF page 60:
https://www.minkowskiinstitute.org/mip/MinkowskiFreemiumMIP2012.pdf#page=60

But this proposal is not accepted today. Maybe, because it would only shift the "i" issue from the spacetime interval (were it does not harm) to everywere else in physics.

If general relativity had not been discovered, then the +--- or -+++ may not have happened. In the late 20th century and even 21st century, there are still books on classical electrodynamics or quantum electrodynamics opting for the $x_4 = ict$, and I wonder why the publishers still accept them.

 

[Sagittarius A-Star]

I know, but every time I have to remember that even though it's the same symbol at the denominator of a second derivative, for instance, in the SR context has another meaning.

I think, the sign of $ds^2$ is not a property of nature, because it is an artifact of the convention, which of the two signatures (+---) or (-+++) you have selected for your given scenario. That means, if you calculate a realistic scenario (for example without tachyons) and calculate only measurable physical quantities, like temporal interval or spatial distance, then the $ds^2$, which might be negative, will automatically disappear from your equations before the end of your calculation. See for example the calculation in posting #3.

 

[PeterDonis]

I think, the sign of $ds^2$ is not a property of nature

More precisely, which kind of interval (spacelike or timelike) you pick to have a positive $ds^2$ and which one you pick to have a negative $ds^2$ is not a property of nature, it's a matter of which signature convention you choose. But the fact that there are intervals with both positive and negative $ds^2$ is a property of nature; there is no way to have only positive (or only negative) $ds^2$ in spacetime by any choice of convention.

 

[PeterDonis]

if you calculate a realistic scenario (for example without tachyons) and calculate only measurable physical quantities, like temporal interval or spatial distance, then the $ds^2$, which might be negative, will automatically disappear from your equations before the end of your calculation.

More precisely, if you know what kind of interval you are dealing with (spacelike or timelike), you can always adjust things during the calculation to avoid having to take the square root of a negative number. However, that doesn't change the fact that spacelike and timelike intervals inherently have opposite signs for $ds^2$. Those opposite signs reflect a fundamental physical difference between those types of intervals.

 

[Sagittarius A-Star]

Ok, Here's something a little more recent

Richard Feynman proposed, that the spacetime interval can be imaginary ...

the interval squared would be negative and we would have an imaginary interval, the square root of a negative number. Intervals can be either real or imaginary in the theory.

Source:
https://www.feynmanlectures.caltech.edu/I_17.html

... and Wolfgang Rindler proposed the opposite:

We shall regard it, as we justify below, as the square of a vector – the "squared displacement" – rather than as the square of a possibly complex number.

Source:
http://www.scholarpedia.org/article/Special_relativity:_mechanics

 

[vanhees71]

H. Minkowski also proposed to shift the "i" issue from the spacetime interval to the unit system by defining the ratio between the unit of distance and the unit of time to be imaginary:
$$3 \cdot 10^5 \ km = \sqrt{-1} \ seconds$$
Source: same link, page 50, PDF page 60:
https://www.minkowskiinstitute.org/mip/MinkowskiFreemiumMIP2012.pdf#page=60

But this proposal is not accepted today. Maybe, because it would only shift the "i" issue from the spacetime interval (were it does not harm) to everywere else in physics.

I hope, we don't need to discuss the infamous $\mathrm{i} c t$ convention. It's a disease ;-)).