Negative Distance Metric: Elapsed Time & Space-Time Interval

[Paige_Turner]

DISTANCE METRIC, 2D:
$$d = \sqrt{x^2 + y^2}$$

DISTANCE METRIC, 3D:
$$d = \sqrt{x^2 + y^2 + z^2}$$

DISTANCE METRIC, 4D:
$$d = \sqrt{x^2 + y^2 + z^2 - ct^2}$$
You SUBTRACT elapsed time from the spatial distance.

Yes, elapsed time is called "imaginary distance," and vice versa. However:

Assuming the space distance x=y=z=0, then absolute distance (the interval) will be purely imaginary. But only the final result is imaginary. The expression SQRT(-ct²) only becomes imaginary in the final SQRT operation. Elapsed time itself (squared) is subtracted from the spatial distance (squared) in ordinary, non-complex arithmetic.

Elapsed time is negative distance.

Time is negative space.

Add to that the fact that gravitational potential is negative mass-energy, and you have a nice zero-energy universe.

NOTE: Anyone who appeals to the dictionary (or Wolfram) to define length and distance as positive, need not reply.

 

[PeterDonis]

DISTANCE METRIC, 4D:

This is the metric for 4D spacetime, not 4D space. There is a perfectly good manifold called 4D Euclidean space in which you sum the squares of the four coordinate intervals to get the squared distance.

$$d = \sqrt{x^2 + y^2 + z^2 - ct^2}$$
You SUBTRACT elapsed time from the spatial distance.

No, you subtract the square of coordinate time from the square of the spatial coordinate distance to get the squared interval. And that's only if you are using the spacelike signature convention; if you are using the timelike signature convention, the signs flip.

Yes, elapsed time is called "imaginary distance," and vice versa.

Not in any reputable textbook that I'm aware of. Can you give a reference?

Assuming the space distance x=y=z=0, then absolute distance (the interval) will be purely imaginary.

No. It will be a time, and that time, as shown by a clock following that particular path through spacetime, is a real number, not an imaginary number.

Elapsed time is negative distance.

Time is negative space.

This is personal speculation, which is off limits here at PF.

 

[PeterDonis]

Anyone who appeals to the dictionary (or Wolfram) to define length and distance as positive, need not reply

The fact that the spacetime interval in relativity does not work like the dictionary definition of "distance" does not mean you get to just make up your own version of how it works.

 

[Ibix]

Assuming the space distance x=y=z=0, then absolute distance (the interval) will be purely imaginary. But only the final result is imaginary.

The measured quantity, be it distance or time, is frequently defined as $\sqrt{|\Delta x^2+\Delta y^2+\Delta z^2-c^2\Delta t^2|}$ precisely to avoid nasty surprises. Particularly since the metric signature (three plus signs and one minus, or three minus signs and one plus) is a matter of convention and sources seem fairly evenly split on which to use.

 

[Paige_Turner]

> This is the metric for 4D spacetime, not 4D space.

Spacetime is what I'm talking about. (Obviously?)

> There is a perfectly good manifold called 4D Euclidean space in which you sum the squares of the four coordinate intervals to get the squared distance.

That's not spacetime. It has the wrong metric sig.

You might be joking or sarcastic. I can't tell because I'm autistic, so I may not be responding properly.

> No, you subtract the square of coordinate time from the square of the spatial coordinate distance to get the squared interval.

I was using simple informal terms. Of COURSE you square the "coordinate distance." God!

> And that's only if you are using the spacelike signature convention; if you are using the timelike signature convention, the signs flip.

Yeah, but so what? The sign convention in SR intervals is the same either way. It's completely irrelevant. Why are you bringing this up?

> Not in any reputable textbook that I'm aware of. Can you give a reference?

No I can't. Minkowski called it "imaginary time" because of the i in ict. I remember someone saying the whole statement, but I can't find it. However, see this (on this site). What you call it doesn't matter.
Note that "space is imaginary time" is the same as "time is imaginary space," because you can multiply both sides by i.

> No. It will be a time, and that time, as shown by a clock following that particular path through spacetime, is a real number, not an imaginary number.

it is a real time, but in distance units, it is negative spatial distance, per the simple interval equation. I mean, the equation is pretty straightforward.

> This is personal speculation, which is off limits here at PF.

Don't say that kind of hostile thing, please. I'm not asserting a crackpot new "theory" like in [Link removed by Mentor]. I'm just trying to understand complex spacetime and I wanted to clarify the meaning of the interval equation.
In fact, I still do.

 

[PeterDonis]

You might be joking or sarcastic.

No, neither. I'm being quite serious. See further comments below.

Minkowski called it "imaginary time" because of the i in ict.

You're not using "ict". If you use "ict" the metric is not the one you wrote down; it's the 4-D Euclidean metric (signs of all terms are positive). However, this metric is also no longer the metric of spacetime, it's the metric of 4-D Euclidean space.

It is true that this mathematical trick was used in the early special relativity literature (the first few decades after Minkowski's 1907 paper on spacetime, IIRC), but the trick has fallen out of favor because, as I note below, you can't change the physics by doing mathematical manipulations. Also because, as explained in some classic General Relativity textbooks like Misner, Thorne, and Wheeler, the "ict" mathematical trick no longer works when spacetime is curved, i.e., when gravity is significant.

The "ict" trick is also used sometimes in quantum field theory (where it goes by the name "Wick rotation"), based on the idea that the 4-D Euclidean space you get with "ict" is an analytic extension of ordinary spacetime, so you can "rotate" over to this 4-D Euclidean space, do some kind of mathematical manipulation, and then "rotate" back to ordinary spacetime and obtain some useful result. None of this justifies the claim that the "ict" thing describes ordinary spacetime.

Note that "space is imaginary time" is the same as "time is imaginary space," because you can multiply both sides by i.

Mathematically, you can use "ict" (or, I suppose, if you used the timelike signature convention to start, you could use "ix / c" and "iy / c" and "iz / c", i.e., multiply the "space" coordinates by $i$ instead of the "time" one, although I have never seen this done in any literature), but that does not mean that physically time is imaginary space (or space is imaginary time). You can't change the physics by doing mathematical tricks.

Also, your original claim, the one that I called personal speculation, was not "time is imaginary space", it was "time is negative space". (And also the claim that "time is negative space" is somehow related to the "zero energy universe" concept.) Those claims are personal speculation, and have nothing to do with the "ict" concept or the idea, based on "ict", that "time is imaginary space".

Don't say that kind of hostile thing, please.

Informing you of the rules here at PF is not being hostile. It is a necessary part of my job as a moderator.

I'm just trying to understand complex spacetime and I wanted to clarify the meaning of the interval equation.
In fact, I still do.

Which is fine, but the way to do that is to ask questions, not to make declarative claims. The only substantive question you have asked so far is the one in the title of this thread, and the answer to that question is simple: no.

Do you have another question?

 

[PeterDonis]

The sign convention in SR intervals is the same either way.

No, it isn't. If you are using the spacelike signature convention (the one you use in your OP, where the "space" squared coordinate differentials are positive and the "time" squared coordinate differential is negative), then a positive squared interval is spacelike and a negative squared interval is timelike. If you are using the timelike signature convention (where it's the "time" squared coordinate differential that's positive), then a positive squared interval is timelike and a negative squared interval is spacelike.

Since none of this changes the actual physics, it should be clear that there is no physical significance to the particular signs that timelike and spacelike squared intervals have; it's purely a matter of our choice of signature convention.

 

[PeterDonis]

it should be clear that there is no physical significance to the particular signs that timelike and spacelike squared intervals have; it's purely a matter of our choice of signature convention.

To be clear, the fact that spacelike and timelike squared intervals have opposite signs does have physical significance: it is a representation in the math of the fact that we measure timelike intervals with clocks and spacelike intervals with rulers. In other words, the two types of intervals are physically different things, measured in physically different ways.

 

[Paige_Turner]

> there is no physical significance to the signs; it's just a choice of signature convention.

How does that differ from what I said: "The sign convention in SR intervals is the same either way."

So I agree with you, right?

I'm always afraid to ask physics questions because I'm autistic and i ask them impolitely or something. Or maybe I'm too eager and it looks weird for a grownup. People have even gotten mad at MATH questions.

> use PF to ask questions, not to make declarative claims.

I asserted, "time is negative distance" as a rhetorical way to restate my question:

"I think time looks like negative distance and here's why. I'm sure this is wrong because nobody else calls it that, so could you please resolve this contradiction because I can't."

See, when autistics talk to normals, they have to bend over backwards to reassure them that you're not insulting or ridiculing them, because normals are obsessed with that. BUT when I ask long, apologetic questions like the above, I'm dismissed as weird and ignored. That would be okay, but nobody nobody ever explains the answers.

> The only substantive question you have asked so far is the one in the title using "ict".

Weyl rotation is interesting and clever and i just read all about it (ty). But i don't WANT to use ict. I want to use the form:

d² = x² + y² + z² - (ct)²

which looks like :

d² = x² + y² + z² - w²

In which case, whatever units x,y,z are in, w is a negative number of those same units. But w is just elapsed time expressed as real distance. If events are the same distance away in space and time, then they are on the null cone, and the invariant distance is zero.

HOWEVER, if the spatial distance (squared) is less than ct (squared), then the the invariant distance is negative.

Umm... right?

I'm sorry if I insulted you or your forum; I just wanted to know why the above isn't correct, because I noticed it 20 years ago and they always get angry because "distance" is defined as positive, delete my questions, and ban me.

When i registered, i knew that would happen here, too, and it's started. But that's not important. I just want to understand what's going on around me, because I never do. Ask anyone. And "what's around me" is the universe.

 

[Ibix]

How does that differ from what I said: "The sign convention in SR intervals is the same either way."

There are two sign conventions. You can write $\Delta s^2=c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2$ in which case timelike intervals have positive $\Delta s^2$ and spacelike intervals have negative $\Delta s^2$. Or you can write $\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2-c^2\Delta t^2$ in which case they are the other way around. This does not seem to be "the same either way".

I'm always afraid to ask physics questions because I'm autistic and i ask them impolitely or something.

I think the main thing is that you need to take criticism with better grace. "You SUBTRACT elapsed time from the spatial distance", which is what you wrote, is not an informal way of talking about the coordinate differences squared in the metric. The squaring is important, and responding to comments about that with "of course I didn't mean what I actually wrote" is silly.

I suggest doing your best to write what you actually mean, not taking shortcuts that may change what you are saying. I also suggest that you need to accept that people won't always understand what you are asking and you may need to go back and forth a few times to get your answer, and this is not necessarily either criticism or attack.

 

[Paige_Turner]

[okay, I'm out of bounds in asking this, so please ignore it. It's just important to me]

> I think the main thing is that you need to take criticism with better grace.

Criticism is perfectly okay. I strip off any normal-people malevolence and take it seriously. But sadly, I can't generate "grace." I literally don't know what it is. To me, statements are just true or false. I can't even see the envelope of emotions that normals embed messages in--much less encrypt them like that myself.

> "You SUBTRACT elapsed time from the spatial distance"is not an informal way of talking about the coordinate differences squared in the metric.

I have no idea why this is true. Was capitalization too emphatic? Would this have been okay:

"One subtracts elapsed time from the spatial distance."

If not, how would you have phrased it politely? Because to me, those two statements are absolutely identical.

> responding to comments about that with "of course I didn't mean what I actually wrote" is silly.

...Yet when I explain what I mean precisely and without emotion, I'm either a pedantic, geeky joke, or my comments are so littered with explanations and footnotes that nobody even reads them.

This isn't a silly joke to me. Dealing with normals is so booby-trapped and frightening and dangerous that I walked off my last job because of it (where I had won awards) and moved into a homeless cave in the woods.

 

[Ibix]

I have no idea why this is true. Was capitalization too emphatic? Would this have been okay:

Because the squaring of the coordinate differences is important. It makes the difference between the (cumbersome and outdated but not wrong) $ict$ convention and the title of this thread (to which the answer is no).

To me, statements are just true or false.

Right. And "[y]ou SUBTRACT elapsed time from the spatial distance" is false. It's the square of the elapsed time and the square of the spatial distance.

Yet when I explain what I mean precisely and without emotion, I'm either a pedantic, geeky joke, or my comments are so littered with explanations and footnotes that nobody even reads them.

None of us always explain ourselves perfectly the first time. We all have to repeat things with more detail or less. All you need to do is, when someone says something you wrote is wrong, either say "yes, you're right" if you already knew that, or think about it and maybe ask more questions if you didn't. If we ask you to explain more, explain more. If we ask for a shorter question, write a shorter form. Say if you don't understand something we ask.

None of us needs sugar coated language. We do expect you to remember that all we know about you is what you write, and if you write something wrong we don't know if that's because you don't know it or because you wrote it wrong, and we need to correct that before we can get anywhere. Eventually we'll come to an understanding.

 

[Paige_Turner]

Thank you for helping me more than you have to.

 

[PeterDonis]

How does that differ from what I said: "The sign convention in SR intervals is the same either way."

I don't know. I'm not sure what you meant by what you said, so I was trying to clarify what the correct statement is. If you agree with what I said, that's fine.

I asserted, "time is negative distance" as a rhetorical way to restate my question:

"I think time looks like negative distance and here's why. I'm sure this is wrong because nobody else calls it that, so could you please resolve this contradiction because I can't."

It would have been better to just ask the question (what you put in quotes here) straight out.

when autistics talk to normals, they have to bend over backwards to reassure them that you're not insulting or ridiculing them, because normals are obsessed with that.

First, I'm not a "normal". I don't need to be reassured. I would prefer that you just ask the exact question you want an answer to.

Second, if you are unsure how people are going to react to what you say (and this is true whether you are autistic, "normal", or anything else), the best way to compensate for that, at least when asking technical questions like what we are discussing here, is to be more literal, not less. The objective here is for us to help you understand something about physics. Muddying the lines of communication for irrelevant "social" reasons does not help with that. You're better off just saying exactly what you mean. After all, you're saying things about physics, not about anyone personally. As a moderator, if some other poster said they were insulted by something you said about physics, I would be chastising them, not you.

Weyl rotation is interesting and clever and i just read all about it (ty). But i don't WANT to use ict.

That's good, because you don't need to.

w is just elapsed time expressed as real distance.

Not quite. It's coordinate time (which, if the observer whose interval you are calculating is moving in the given frame, is not the same as the observer's elapsed time, which is the timelike interval itself) measured in the same units as distance. That doesn't make time the same as distance. It just means they're measured in the same units so that we don't have factors of $c$ cluttering up the formulas.

HOWEVER, if the spatial distance (squared) is less than ct (squared), then the the invariant distance is negative.

No, the invariant squared interval is negative, which means, with the signature convention you have chosen, that it is timelike.

Perhaps it might help to correct the implicit rules you appear to be using to deal with intervals. Assume that you have chosen the spacelike signature convention, so the squared interval is $ds^2 = dx^2 + dy^2 + dz^2 - dt^2$. Then the rules for dealing with intervals are:

(1) If the squared interval is positive, then (a) you take its positive square root, (b) which you interpret as a spatial distance, i.e., something measured by a ruler.

(2) If the squared interval is negative, then (a) you take the positive square root of its absolute value, (b) which you interpret as an elapsed time (for an observer following the timelike path through spacetime that the interval describes), i.e., something measured by a clock.

(3) If the squared interval is zero, then (a) the interval itself is obviously zero, (b) which you interpret as a null path (path of a light ray).

Note that these rules are rules of physical interpretation, not math; they tell you how to match up the math with physical observables (like ruler measurements of distance and clock measurements of time).

(Notice, btw, that I had to say "take the positive square root" in the above--which addresses an issue you apparently didn't consider, the fact that there are two square roots and you have to pick which one to use.)

I'm sorry if I insulted you or your forum

You didn't.

 

[PeterDonis]

It's the square of the elapsed time and the square of the spatial distance.

No, it isn't. It's the squares of coordinate differentials. That makes a difference. See below.

"One subtracts elapsed time from the spatial distance."

No. One subtracts the coordinate time differential from the coordinate spatial differentials. Coordinate time differential is not the same as elapsed time, and coordinate spatial differentials are not the same as spatial distance.

In other words, if you are interpreting $dx^2$ as "spatial distance in the x direction", and similarly for $dy^2$ and $dz^2$, and if you are interpreting $dt^2$ as "elapsed time", you are incorrect. Those are just coordinate differentials, not actual measurements.

Let me elaborate on this a bit. Let's leave out the $y$ and $z$ directions for simplicity and just consider motion in the x direction. Suppose we have a straight line in spacetime that goes from $(x,t)=(0,0)$ to $(x,t)=(1,2)$. We compute the squared interval as $ds^2=dx^2−dt^2=1^2−2^2=−3$. What is this telling us? Per my previous post, it is telling us (a) that we have an interval of $\sqrt{3}$ (b) which we interpret as an elapsed time (proper time) on a clock that follows the given path through spacetime, i.e., a clock whose worldline is a straight line from $(0,0)$ to $(1,2)$.

Notice that the above, in itself, tells us nothing about any spatial distance. If we want to interpret $dx$ as a spatial distance, we have to add more observers to our scenario: we need to have two observers, one at $x=0$ and one at $x=1$, who are at rest relative to each other and in the frame in which we are doing our calculations. Then we could say that $dx=1$ is the spatial distance between those two observers, because if we take both of those observers at the same coordinate time, say $t=0$, then the squared interval between them is $ds^2=dx^2−dt^2=1^2−0^2=1$, which, per my previous post, we interpret (a) as an interval of $1$ (b) which is a spatial distance, i.e., the length measured on a ruler along the spacelike line from $(0,0)$ to $(1,0)$. Then, since the two observers are at rest in our frame, we could say that this is also the spatial distance that the clock travels; but note that the clock does not go from $(0,0)$ to $(1,0)$, it goes from $(0,0)$ to $(1,2)$. So interpreting $dx$ as a spatial distance requires us to look at a different interval (a different line through spacetime) than the one the clock is traveling on.

Similar remarks apply if we want to interpret $dt$, the coordinate time difference, as an "elapsed time"; it is not the elapsed time for the clock itself (that is $\sqrt{3}$, as we have already shown), and interpreting it as "elapsed time" for either of our other two observers (the one at rest at $x=0$ or the one at rest at $x=1$) requires looking at different spacetime intervals--different lines through spacetime--than the one the clock is traveling on. (I'll leave you to think about which lines through spacetime those are, and how to calculate their intervals to get an elapsed time of $2$.)

I agree that, once you have gone through the above reasoning, there is a strong temptation to think about the coordinate differentials $dx$ and $dt$ as "spatial distance" and "elapsed time" anyway, because it seems innocuous--after all, they turn out to be the same, right? But it isn't innocuous. Aside from the fact that doing that seems to be causing you confusion, it also is a bad habit to get into because all of this nice, simple arrangement where coordinate intervals end up being exactly the same as spatial distances or elapsed times in some frame stops working when you add gravity to the mix and have to use curved spacetime (i.e., general relativity) instead of flat spacetime (which is what we're dealing with here). In a curved spacetime, in general, coordinate differentials have no physical meaning, and thinking of them as having physical meaning causes all sorts of problems. (Note that we can even find coordinates in flat spacetime that raise similar issues--for example, null coordinates or Rindler coordinates.)

Also, in curved spacetime, the metric can get much more complex than just putting a minus sign in front of $dt^2$. So asking "why" the metric takes a particular form gets even more problematic--unless you're willing to accept the only real answer, which is "because that's what is required to correctly describe the spacetime geometry". We don't start with coordinates; we start with a spacetime geometry, and we have to figure out how to describe it using coordinates. The minus sign in the metric of flat Minkowski spacetime is there because we've found that it's necessary to correctly describe the geometry of flat Minkowski spacetime using Cartesian $x$, $y$, $z$, $t$ coordinates. That's really all there is to it.