# Understanding ct: Time vs Distance on Spacetime Diagrams

• DocZaius
In summary, the concept of a "ct" axis in spacetime diagrams involves the use of units to maintain dimensional consistency in 4-vectors. However, this approach is not necessary in the framework of special relativity, as the universe is considered a single, cohesive entity of spacetime. The idea of a distinct "time" dimension is a nonrelativistic mode of thought, and all dimensions should be considered equally "spacetime".
DocZaius
ct axis??

Now that I am looking at spacetime diagrams that involve the speed of light, I am seeing the vertical axis as "ct". Since "c" is meters/second and "t" is seconds then wouldn't ct be ((meters/seconds)*seconds) and end up being METERS? Why would the time axis be in meters?

I would think that I would want my time axis to be just in time (not ct), and simply to be told that the x to t scale is 299,792,458 meters for every second. In other words, every unit of the grid horizontally is 299,792,458 meters and every unit of the grid vertically is 1 second. This keeps the vertical axis as purely time and the horizontal axis as purely distance.

So in summary: doesn't "ct" end up being a measure of distance, not time? And if so, why is it on the "time" axis?

Thanks.

If your "time" axis is ct then you are guaranteed that light goes at a 45º angle regardless of your units for t and x. Also, when you are using 4-vectors they are always specified as (ct,x,y,z) for dimensional consistency. Also, it makes it clear that you are trying to think about relativity geometrically.

Tell me where I go wrong please:

1) Doesn't ct mean c * t?
2) Then doesn't that mean (meters/seconds) * seconds?
3) Doesn't that come out to be just meters?
4) Wouldn't a ct axis thus be in meters?

Yes, in SI units ct is meters. But it is still measuring time.

Think about it as the reverse of this common thing: Someone may ask you where you live and you could reasonably respond "I live one hour south of Dallas". You are describing a spatial distance, but you are using units of time with an implied conversion factor of some speed (e.g. 70 mph).

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Thanks for that info, it is helpful.

So where does the x-axis come in?

In other words if the ct axis says "I live one hour south of Dallas (with an implied speed)" then what statement does the x-axis make?

I've thought some more about this, and I've come to the (possibly wrong) conclusion that the vertical axis is "LIGHT-METERS" (which ends up being a measurement of time, much as light-seconds is a measurement of distance) and the horizontal axis is "METERS". This keeps the vertical axis measuring the time dimension and the horizontal axis measuring a spatial dimension.

A light meter is the amount of time it takes light to travel one meter.

Would you say this is a reasonable way of looking at the axes?

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Sure, that is reasonable. Although it is not necessary to specify. It could be "light feet" or "light miles" or whatever. In any case the worldline of a photon is at a nice convenient 45º angle.

Right, I was using the SI units as an example.

Thanks.

DocZaius said:
Why would the time axis be in meters?

Because it's needed for what DaleSpam calls "dimensional consistency" of 4-vectors.

The separation of a 4-vector is $$\LARGE (ct)^2\,-\,x^2\,-\,y^2\,-\,z^2$$, so everything in that expression has to be in the same units - in this case, metres-squared.

(In practice, we actually tend to express everything as time-squared rather than distance-squared: in space-travel, time is measured in years, and distance is measured in light-years, which are really just years. Think about it!)

You can't subtract a length-squared from a temperature-squared, and similarly you can't subtract a length-squared from a time-squared if length and time are different! You can only add or subtract things of the same type.

Time and space wouldn't be interchangeable if they were different.

"Parable of the Surveyors"

Thanks, robphy!

DocZaius said:
This keeps the vertical axis measuring the time dimension and the horizontal axis measuring a spatial dimension.
...
Would you say this is a reasonable way of looking at the axes?
No; that is an inherently nonrelativistic mode of thought. In SR, the universe is not decomposed into space and time, but is instead a single, cohesive whole: spacetime. An individual direction can be described as being space-like or time-like (or light-like), but there simply is not a (physical) separation of the universe into spatial and temporal components.

This line of thinking even has problems with Galilean relativity -- while in that theory there does exist a universal temporal coordinate, space can mix with time through a shear transformation (i.e. a Galilean boost, or a change of inertial reference frame), so even here we cannot decompose the universe into a spatial and temporal components.

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Hurkyl said:
No; that is an inherently nonrelativistic mode of thought. In SR, the universe is not decomposed into space and time, but is instead a single, cohesive whole: spacetime. An individual direction can be described as being space-like or time-like (or light-like), but there simply is not a (physical) separation of the universe into spatial and temporal components.This line of thinking even has problems with Galilean relativity -- while in that theory there does exist a universal temporal coordinate, space can mix with time through a shear transformation (i.e. a Galilean boost, or a change of inertial reference frame), so even here we cannot decompose the universe into a spatial and temporal components.

So you're saying that the 4th dimension is identical to the first 3 dimensions, since it is also spacetime?

I thought the reason for the 4th dimension was to allow time in the picture. But you seem to say that time was ALREADY in the picture in the first 3 dimensions. So then why is the 4th dimension needed?

Why not just say: "There are 3 dimensions: spacetime, spacetime, and spacetime" ?

Everytime I see a reference to the 4th dimension in diagrams and relativity discussions, it is very specific to "time". Are all these people in error to make that dimension so distinctively "timelike"? Would it have been more proper to give all the dimensions the same amount of "spacetimeness" ?

Thanks for any help...

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Hurkyl said:
space can mix with time through a shear transformation (i.e. a Galilean boost, or a change of inertial reference frame).

I don't follow that - can you explain?

DocZaius said:
Everytime I see a reference to the 4th dimension in diagrams and relativity discussions, it is very specific to "time". Are all these people in error to make that dimension so distinctively "timelike"? Would it have been more proper to give all the dimensions the same amount of "spacetimeness" ?

You're right - time is always specifically singled out as different to space.

Although they are measured in the same units.

It's like the directions on a cylinder or a torus: loosely speaking, they have a "short" direction and a "long" direction, but they're both measured in the same units.

Similarly, space directions are "positive-squared" while time directions are "negative-squared", and so it's very important to distinguish between them.

tiny-tim said:
I don't follow that - can you explain?
Suppose at a particular time, I choose a particular point P. After time passes (say, one second), does it still make sense to talk about the point P?

In a non-relativistic description of the universe, it does; position is absolute.

In Galilean relativity, it does not. Any system of locating P at future times amounts to selecting a preferred frame of reference.

In special relativity, the question itself is nonsense, since it presumes an absolute notion of time.

Galilean relativity

Oh I see!

You mean t´ = t, x´ = x + vt, and so the new space coordinate x´ is always a combination of the original space and time.

Got it!

So which interpretation of the dimensions of spacetime is most accurate?

1) 3 dimensions: spacetime, spacetime, spacetime
2) 4 dimensions: space, space, space, time
3) 4 dimensions: spacetime, spacetime, spacetime, spacetime
4) 1 dimension: spacetime

If some are ridiculous, please ignore them.

Thanks.

I'll go for choice 2):
4 dimensions: space, space, space, time,​
but with the proviso that they are always measured in the same units.

tiny-tim said:
I'll go for choice 2):
4 dimensions: space, space, space, time,​
but with the proviso that they are always measured in the same units.

Me too. Further:

Position:

$$\vec{P}=(x,y,z,t)$$

Velocity:

$$\vec{V}=\frac{\delta \vec{P}}{\delta t}=(\frac{\delta x}{\delta t},\frac{\delta y}{\delta t},\frac{\delta z}{\delta t},1)$$

Acceleration:

$$\vec{A}=\frac{\delta \vec{V}}{\delta t}=(\frac{\delta^2 x}{\delta t^2},\frac{\delta^2 y}{\delta t^2},\frac{\delta^2 z}{\delta t^2},0)$$

Regards,

Bill

DocZaius said:
So which interpretation of the dimensions of spacetime is most accurate?

1) 3 dimensions: spacetime, spacetime, spacetime
2) 4 dimensions: space, space, space, time
3) 4 dimensions: spacetime, spacetime, spacetime, spacetime
4) 1 dimension: spacetime

If some are ridiculous, please ignore them.

Thanks.
The thing is, there aren't individual things called "dimensions" that you can point to and say "that one's space", "that one's time", et cetera.

(If you're unconvinced of that fact, then consider the two-dimensional surface of the Earth -- what is a dimension in that case? Is North a dimension? East? Northeast? South by Southwest?)

Prerelativistically, there is a way to partially make sense of there being individual dimensions; space-time is decomposed into a one-dimensional time and a three-dimensional space. (algebraically, there is a canonical way to decompose spacetime into $L \times S$, where L is a Euclidean line, and S is three-dimensional Euclidean space)

In Galilean relativity, there is not an intrinsic way to separate space-time into time and space. (e.g. there is no 'preferred reference frame') However, you can still describe space-time as consisting of one-dimensional time, and for each instant of time, a three-dimensional space. (in particular, there is no absolute notion of position -- the notion of position is relative to time)

In special relativity, you cannot even single out an absolute notion of time. You just have a 4-dimensional space-time. (or you might say (3+1)-dimensional, if you were using a certain, more nuanced definition of dimension)

There is a distinction, though, between "time-like" and "space-like" (and light-like) for directions. We have facts like there exist three-dimensional space-like sets, but there cannot exist a two-dimensional timelike set. But among timelike vectors, there is none that you can single out and say "this is a time dimension".

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Hurkyl said:
In special relativity, you cannot even single out an absolute notion of time. You just have a 4-dimensional space-time. (or you might say (3+1)-dimensional, if you were using a certain, more nuanced definition of dimension)

There is a distinction, though, between "time-like" and "space-like" (and light-like) for directions. We have facts like there exist three-dimensional space-like sets, but there cannot exist a two-dimensional timelike set. But among timelike vectors, there is none that you can single out and say "this is a time dimension".

I think this analysis is confusing.

As you say, "time-like" and "space-like" directions are definitely different. All observers agree about which is which.

Any four perpendicular non-light-like directions in space-time will always include one time-like and three space-like directions.

If you use those four directions as your four dimensions t x y and z, then there's no doubt which one should be t: it should be the only time-like one!

In that sense, everyone who calls t x y and z "dimensions" not only can but should call t the time dimension!

It's definitely different from the other three!

The thing is, there aren't individual things called "dimensions" that you can point to and say "that one's space", "that one's time", et cetera.

You can't point to dimensions at all - you can point to (or draw) a direction, but dimension is an abstract concept.

For example, you can point to "a length", but not to "length"!

(If you're unconvinced of that fact, then consider the two-dimensional surface of the Earth -- what is a dimension in that case? Is North a dimension? East? Northeast? South by Southwest?)

No. North is a direction! So are East, Northeast, and South by Southwest.

And they're all space-like. So the dimensions associated with them are space dimensions.

(Though, of course, different map-makers might make different choices as to which pair of directions to make their x and y. But that freedom of choice doesn't mean that there's no such thing as space dimensions!)

And time can be defined as a fourth perpendicular direction - in which case, the associated dimension is a time dimension!

Hurkyl said:
The thing is, there aren't individual things called "dimensions" that you can point to and say "that one's space", "that one's time", et cetera.

I offered the "spacetime, spacetime, spacetime" choices for that very reason. If this space/time separation is inaccurate, then surely regarding each dimension as itself enveloping both space and time (spacetime) would be a more relativistic notion. Please adress the "spacetime, spacetime, spacetime" choices and why they are more wrong than "space, space, space, time" - especially considering that you seem to discourage the latter's separation of the concepts.

Hurkyl said:
In special relativity, you cannot even single out an absolute notion of time. You just have a 4-dimensional space-time. (or you might say (3+1)-dimensional, if you were using a certain, more nuanced definition of dimension)

I'm not sure what you are trying to say. The Minkowski metric clearly assumes a Euclidean relationship (e.g. mutual orthogonality) between the four components of "space-time". However, the Minkowski metric treats "time" as an imaginary component having a factor of "c" to represent the light-like length of a world-line in the same spatial direction. Consider the length of the following vector:

$$\vec{s}=(x,y,z,ict)$$ , where $$i=\sqrt{-1}$$

$$s^2=x^2+y^2+z^2-(ct)^2$$

For anything traveling along a constant 3-space direction at the speed of light, s=0.

If I now write the usual form of the Minkowski metric (as I think I've seen it), I have:

$$ds^2=x^2+y^2+z^2-(ct)^2$$

A non-zero "ds" in the above appears to represent a "differential world-line" that accounts for either a curved trajectory through space (that ultimately results in the x,y,z displacement), or a less than maximal spatial displacement due to sub-luminal velocity (or both). If this "differential world-line" is to make any sense to me, ds must decompose into a negative 4-space velocity (either negative space or negative time, but not both) pointing back to the start of the corresponding "world-line".

If there's a more conventional interpretation of the above, I'd be glad if someone would be so kind as to point me to an appropriate reference.

There is a distinction, though, between "time-like" and "space-like" (and light-like) for directions. We have facts like there exist three-dimensional space-like sets, but there cannot exist a two-dimensional timelike set. But among timelike vectors, there is none that you can single out and say "this is a time dimension".

Do you mean to imply that "time" is not proportional to the vector cross product of a spatial direction (zero time component), and a space-time velocity in the same spatial direction?

Regards,

Bill

@ tiny-tim and Antenna_Guy:

I don't feel it's necessary to hold onto the distinction between time and space. Treat all four as spacetime dimensions.

For example, consider lightcone coordinates, where the usual time coordinate is encoded in the lightcone + and - components along with the usual x coordinate. In this example, holding on to time being one dimension and space being three is not very useful.

Particularly in general relativity, where you are allowed to take arbitrary transformations. Only the signature of the metric must stay -1,1,1,1, everything else can change.

@ DocZaius:

You can perhaps make up your own mind, but I would go for option 3: spacetime is composed of 4 dimensions of the same thing. For some reason, one of those dimensions have a minus sign when we try and measure distances in this space time.

masudr said:
@ tiny-tim and Antenna_Guy:
For example, consider lightcone coordinates, where the usual time coordinate is encoded in the lightcone + and - components along with the usual x coordinate. In this example, holding on to time being one dimension and space being three is not very useful.

I'm not really sure what to make of a light-cone; perhaps you could explain it for me within the context of a photon passing a massive body:

Consider a photon at its' point of closest approach to a massive body. At this point, the velocity vector of the photon is tangential to a sphere about the massive body. The gravitational pull of the massive body bends the path of the photon effectively from a point perpendicular to its' direction of travel.

Question #1: Is the massive body acting from the future, past or present of the photon's "light-cone"?

Question #2: What does the concept of a "light-cone" tell me about the past/future of the photon in this example?

Regards,

Bill

Bill, see this for an introduction to lightcone coordinates.

## 1. What is a spacetime diagram?

A spacetime diagram is a graphical representation of the relationship between time and distance in the theory of relativity. It combines the three dimensions of space with the dimension of time to create a four-dimensional visualization of events.

## 2. How do you plot time and distance on a spacetime diagram?

Time is typically plotted on the vertical axis, while distance is plotted on the horizontal axis. This allows for a visual representation of how time and distance are related in a given event.

## 3. What is the purpose of a spacetime diagram?

The purpose of a spacetime diagram is to help visualize and understand the effects of time dilation and length contraction in the theory of relativity. It also allows for the calculation of spacetime intervals between events.

## 4. How do you interpret a spacetime diagram?

In a spacetime diagram, the diagonal line represents the speed of light, which is constant in all reference frames. The steeper the slope of a line, the faster the object is moving through space. The horizontal lines represent objects at rest in a specific reference frame.

## 5. What is the relationship between time and distance on a spacetime diagram?

The relationship between time and distance on a spacetime diagram is not linear, as it is in traditional graphs. This is because of the effects of time dilation and length contraction, which occur at high speeds or in the presence of strong gravitational fields.

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