Is proper time a vector?

Question

Proper time is identical with the spacetime interval of a timelike movement.

A spacetime interval is the dot product of two vectors and thus a scalar. Proper time however is always pointing exactly in time direction, all space dimensions are 0. Also it may be added or multiplied with scalars.

As outlined in the answer of Ross Millikan, proper time does not take part in Lorentz transforms. That means that it is not a vector within Minkowski vector space. But with its above mentioned characteristics, is there something missing for proper time being a vector?

Or, in case that the physical definition of proper time is too restricted, what else is this vector pointing in time direction with the magnitude of proper time?

Edit: The most confusing fact is the comparability of time (vector) and proper time (scalar?), example: A space traveler returns after 30 years, but he got only 20 years older, thus he "saved" 10 years (substraction vector / scalar??).

Answer 1

Proper time is the dot product of two four-vectors (actually one vector with itself). As such it is a scalar. You can see it is not a part of a vector by the fact that it is not changed by rotations or boosts.

Answer 2

You say:

Proper time however is always pointing exactly in time direction

but this is not so. It is certainly true that in an observer's rest frame the proper time is numerically equal to the coordinate time, but this does not mean that the proper time and the coordinate time are the same. The proper time is still defined by:

$$ d\tau^2 = g_{ab}dx^a dx^b $$

so it is still a scalar. It's just that in the rest frame only $x^0$ (i.e. $dt$) is non-zero so we have:

$$ d\tau^2 = dt^2 $$

In other inertial frames $dx$, $dy$ and $dz$ may not be zero, but $d\tau$ will be the same (because it's invarient under Lorentz transformations) so in general $d\tau$ will not be equal to $dt$.

Answer 3

Is proper time a vector?

Unequivocally, no. Proper time is a scalar, not a vector. From the Wikipedia article "Scalar (physics)":

Examples of scalar quantities in relativity include electric charge, spacetime interval (e.g., proper time and proper length), and invariant mass.

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Proper time however is always pointing exactly in time direction

Proper time, as a scalar, is a number without direction; proper time does not point, period.

This is elementary and, evidently, at the root of your misunderstanding.

Again, from the Wikipedia article "Scalar (physics)":

In physics, a scalar is a one-dimensional physical quantity, i.e. one that can be described by a single real number (sometimes signed, often with units), unlike (or as a special case of) vectors, tensors, etc. which are described by several numbers which characterize magnitude and direction

So, your conception of proper time is flawed and you must give up this idea that proper time points in direction, time or otherwise. It does not.

Answer 4

A timelike vector of unit length can have $(ct)^2-x^2=1$, and this traces out a hyperbola of two sheets symmetric after reflections through the $x$ axis. There are an infinite number of vectors with the proper time of 1, but they point in vastly different directions.

Generally, "time coordinate" and "proper time" are treated as two completely separate things. In the same way, "x coordinate" and "length" are two completely separate things. The x coordinate can be negative or undefined while length is still defined, and length is defined so that it is always positive. It's a scalar, not a vector. Proper time is a scalar, not a vector. It does not "point in the time direction with all space coordinates zero".

If you're still in doubt, try to write down a mathematical definition of proper time, or try to find some way to apply it to a problem. If your definition is like the ones given by John Rennie or the second paragraph of this post, you will find that proper time is a scalar and not a vector. (By the way, when $\tau^2$ is negative, we call $l^2=x^2+y^2+z^2-(ct)^2$ a proper length)

Answer 5

Moonraker: "Edit: [...] comparability" --

How to compare

• the magnitude $s[ \mathcal{E}_{AJ}, \mathcal{E}_{AQ} ]$ of a particular time-like interval between two particular events "$\mathcal{E}$" where the indicated participants met each other; i.e. at one $A$ and $J$ met each other (but not $Q$), and at the other event $A$ and $Q$ met each other (but not $J$) to

• the duration $\tau A[ \circ_J, \circ_Q ]$ of some particular participant $A$ from $A$'s indication at one initial event (the meeting with participant $J$) until $A$'s indication at the other, subsequent final event (the meeting with participant $Q$)

?
That's an important question (which surely has been raised and addressed on this site, too).
And there's a cute, somewhat superficial and decidedly mathematical answer:

The value of the ratio

$$ \tau A[ \circ_J, \circ_Q ] ~ / ~ s[ \mathcal{E}_{AJ}, \mathcal{E}_{AQ} ] $$

is (equal to) the limit of the sum of ratios

$$ \text{Limit}_{ \mathscr{\hat S} \rightarrow \mathscr{A}_{J~Q}; \text{and for successive event pairs } \in (\mathscr{\hat S} \cup \{ \mathcal{E}_{AJ}, \mathcal{E}_{AQ} \}): s[ \mathcal{E}_{A\hat K}, \mathcal{E}_{A\hat P} ] ~ / ~ s[ \mathcal{E}_{AJ}, \mathcal{E}_{AQ} ] \rightarrow 0 } \large{[} \sum_{ \text{successive event pairs } \in (\mathscr{\hat S} \cup \{ \mathcal{E}_{AJ}, \mathcal{E}_{AQ} \} } s[ \mathcal{E}_{A\hat K}, \mathcal{E}_{A\hat P} ] ~ / ~ s[ \mathcal{E}_{AJ}, \mathcal{E}_{AQ} ] \large{]}, $$

where

• set $\mathscr{A}_{JQ}$ is the set of all events in which $A$ took part from (including) the initial event $\mathcal{E}_{AJ}$ of having met $J$ until (including) the final event $\mathcal{E}_{AQ}$ of having met $Q$,

• set $\mathscr{\hat S}$ is a (any variable) subset of $\mathscr{A}_{JQ}$ consisting of discrete successive events (in which $A$ took part; such as $\mathcal{E}_{A\hat K}$ for any suitable (variable) participant $\hat K$ and $\mathcal{E}_{A\hat P}$ for any suitable (variable) participant $\hat P$),

• and the limit (if it exists, for the particular participant $A$, the particular initial event $\mathcal{E}_{AJ}$ and the particular final event $\mathcal{E}_{AQ}$) is taken as ever more (discrete successive) events of $\mathscr{A}_{JQ}$ are included in $\mathscr{\hat S}$, and

• all ratios $s[ \mathcal{E}_{A\hat K}, \mathcal{E}_{A\hat P} ] ~ / ~ s[ \mathcal{E}_{AJ}, \mathcal{E}_{AQ} ]$ between the magnitude of an interval between any two consecutive events in set $\mathscr{\hat S} \cup \{ \mathcal{E}_{AJ}, \mathcal{E}_{AQ} \}$ and the magnitude of the interval between initial and final event approaches $0$.

This limit (if it exists) constitutes a Riemann integral and may accordingly be written as

$$ \tau A[ \circ_J, \circ_Q ] ~ / ~ s[ \mathcal{E}_{AJ}, \mathcal{E}_{AQ} ] := \int_{\mathscr{A}_{J~Q}} ds_{JQ}.$$

So far, so good. (Hopefully.) However, it remains the task to compare magnitudes of (time-like) intervals to each other in the first place; i.e. the question should be addressed how the real number values of ratios $$s[ \mathcal{E}_{A\hat K}, \mathcal{E}_{A\hat P} ] ~ / ~ s[ \mathcal{E}_{AJ}, \mathcal{E}_{AQ} ]$$ ought to be determined, by geometrical physical measurement.

This primary question (of physics) must be addressed without presuming or requiring results of comparisons discussed above, of course.
Not surprisingly, that's quite difficult in general; some basics approach to an answer is sketched for instance in (my answer to the question) "Deriving formula for time dilation".

(That question as well as my indicated answer presume and require the notion of certain participants having pairwise been and remained "at rest to each other"; which therefore must in turn be determined without presuming or requiring results of comparisons discussed above.)

Answer 6

Let's measure our vectors in meters, just to pick a convention (afterwards I'll do it in seconds so that you can see that it doesn't affect the basic ideas).

Given a point in spacetime $(ct_1,x_1,y_1,z_1)$ and another point $(ct_2,x_2,y_2,z_2)$, we can easily construct three things.

1) A vector $\vec{v}=(c(t_2-t_1),x_2-x_1,y_2-y_1,z_2-z_1)$ that points from one to the other.

2) The proper length $l=\sqrt{c^2(t_2-t_1)^2-(x_2-x_1)^2-(y_2-y_1)^2-(z_2-z_1)^2}$ of that timelike vector $\vec{v}$.

3) And finally, the unit vector $\vec{u}=\vec{v}/l$, which equals $\frac{(c(t_2-t_1),x_2-x_1,y_2-y_1,z_2-z_1)}{\sqrt{c^2(t_2-t_1)^2-(x_2-x_1)^2-(y_2-y_1)^2-(z_2-z_1)^2}}$.

As numbers (scalars) and vectors (geometric displacements), everyone will agree. The vectors might appear as different quadruples of numbers to different observers (imagine picking a different direction to call your $x$-axis, the geometric entity is the same displacement, but you get different numbers).

Now if you didn't want to measure in meters you can do it again, dividing everything by $c$, except it turns out $u$ will be the same.

1) A vector $\vec{V}=(t_2-t_1,(x_2-x_1)/c,(y_2-y_1)/c,(z_2-z_1)/c)$ that points from one to the other.

2) The proper length $L=\sqrt{(t_2-t_1)^2-(x_2-x_1)^2/c^2-(y_2-y_1)^2/c^2-(z_2-z_1)^2/c^2}$ of that timelike vector $\vec{V}$.

3) And finally, the unit vector $\vec{U}=\vec{V}/L=\vec{u}$, which again equals $\frac{(c(t_2-t_1),x_2-x_1,y_2-y_1,z_2-z_1)}{\sqrt{c^2(t_2-t_1)^2-(x_2-x_1)^2-(y_2-y_1)^2-(z_2-z_1)^2}}$.

So you have a vector $\vec{v}$ between two spacetime points, as a vector it has a length $l$, and a corresponding unit vector $\vec{u}=\vec{v}/l$ that keeps track of the direction in spacetime, but ignores how much dispalcement (or even which units you use, just the direction in spacetime).

OK, so now lets get to your question. Given two points we have the spacetime displacement $\vec{v}$, the unit displacement $\vec{u}$ that tells us which direction in spacetime to go (but has no informative length or even units), and the length $l$ telling us how far in spacetime to go (and has units of length or time). When someone says proper time, they are referring to the length of the timelike vector, maybe in units of time instead of length.

A common reason people say proper length for spacelike entities and proper time for timelike entities in textbooks isn't because they care a great deal about the units of the answer (it's just a measure of length) but because the squared length $l^2=c^2(t_2-t_1)^2-(x_2-x_1)^2-(y_2-y_1)^2-(z_2-z_1)^2$ is what is easy to compute and for a timelike spacetime displacement $c^2(t_2-t_1)^2-(x_2-x_1)^2-(y_2-y_1)^2-(z_2-z_1)^2$ is positive, whereas for a spacelike displacement $-c^2(t_2-t_1)^2+(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2$ is positive. We like to have square roots of positive numbers, so people like to have formulas like $\tau=c^-1\sqrt{c^2(t_2-t_1)^2-(x_2-x_1)^2-(y_2-y_1)^2-(z_2-z_1)^2}$ and $l=\sqrt{-c^2(t_2-t_1)^2+(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$. The only real reason to have two formulas is because then you are taking the square root of a positive number. The idea for either case is the same: the displacement has a magnitude (proper length or proper time) and a direction (unit vector in spacetime), their product gives you the displacement.

Just like in three-space, a vector $(\Delta x,\Delta y,\Delta z)$, has a magnitude, $r=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}$, and a unit vector (for direction) $((\Delta x)/r,(\Delta y)/r,(\Delta z)/r)$.

Edit: I guess I didn't actually answer the questions.

First question: Is there something missing for proper time being a vector?

Yes, proper time is a magnitude, not a vector, it is just a length in spacetime, it by itself has no direction, there are many directions you could travel all with the same length/magnitude.

Second question: What else is this vector pointing in time direction with the magnitude of proper time?

The vector pointing from one event to the other will look like $(\tau,0,0,0)$ in the frame that inertially moves from one event to the other. But the vector $(\tau,0,0,0)$ is different than the scalar $\tau$. It's like if you walked north a distance of five meters, the distance is 5, everybody agrees. And to you the space vector might be $(5,0,0)$ but what if someone else like to put east first and north second? Then they will say that the vector is $(0,5,0)$. A vector has to tell you the direction. If all you tell me is you walked 5 units I can't tell if you walked $(5,0,0)$, $(0,5,0)$, $(0,0,5)$, or even $(3,4,0)$. The vector tells you the direction and the magnitute, but only when you say what the 3 (or 4) numbers mean first.

Answer 7

Proper time is a directed scalar. As of this writing, there is no mathematical terminology or analysis technique that does justice to defining it, including and especially as proper time was defined by Minkowski, hyperbolic / rotationally bound to something else defined as space, which is the same directed scalar.

For a directed virtual or real EM wave of any wavelength (think of a very long real one, for example), a great many energy transfer events may take place from crest to crest. Hence, an EM wave is not fundamentally related to time, even though from a 19th century perspective, it was considered the fastest anything could move.

Instruments that measure time only accomplish their function by comparing the proportional speeds at which disparate events occur, and take no account of either time dilation or proper time in order to do so. The faster the relative speed upon which a timepiece is based is, the better. Currently, quantum entanglement, not the propagation of an EM wave in free space, would yield an improved standard of time. To realize this fact, one need only imagine trying to gauge the rate at which entanglement states flip using a clock based on the propagation of an EM wave, compared to the reverse.

Because the bound energy we refer to as matter is required in order to observe entanglement, it makes sense that the pairing of EM waves needed to create matter-antimatter pairs results in internal relative rotational or other modes of propagation speeds in excess of that we observe of EM waves in a vacuum. It also makes sense that it is time dilation that renders a form of energy that persists as matter.

Proper time for the bound energy that is matter depends on the time dilation observed at their geometric centers being lower than at their outer boundaries.

A discussion of proper time for an EM wave that does not consider what proper time means for matter is an incomplete description, whether there is math to support the description or not.