# Spacetime interval

## Main Question or Discussion Point

Hello everyone
Please help me understand why there is a minus in the spacetime interval formula and not a plus

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Staff Emeritus
2019 Award
Let's turn it around - why do you think there shouldn't be one there?

pervect
Staff Emeritus
The constancy of the speed of light implies that if the Lorentz interval is zero for one observer, it's zero for all. This works only because of the minus sign - it's basically a restatement of the principle of the constancy of the speed of light.

The stronger statement that the Lorentz interval is constant for all observers implies the above, for if it's constant and zero for one observer, it's zero for all. It turns out that the stronger formulation is true, but I'm not aware of any easy to explain reason why the stronger formulation turns out to be true.

Hopefully, this provides some insight into the motivation of why we need a minus sign, even if it doesn't totally explain it.

The negative sign implies that there is something a little different about the time dimension as opposed to the spatial dimensions. In particular you cannot rotate about one of the spatial axis and point backwards in the time dimension but you can point backwards in a spatial dimension.

Chestermiller
Mentor
The sign is negative because the fundamental geometry of 4D spacetime is non-Euclidean, and, unlike Euclidean geometry where the Pythagorean metric (featuring a plus sign) applies, in non-Euclidean spacetime, the Minkowski metric (featuring the negative sign) applies.

ZapperZ
Staff Emeritus
I am with Vanadium. Would you be equally puzzled if it was a "+" sign instead? This question appears to be based simply on "aesthetics".

Zz.

robphy
Homework Helper
Gold Member
In a plane, if you ask a bunch of surveyors initially at the origin to travel 1 mi (according to each surveyor's odometer) in all possible directions, their endpoints trace out a circle [in Euclidean Geometry]
$$(1\ \rm{mi})^2=(\Delta x)^2+(\Delta y)^2$$

In a position-vs-time graph, if you ask a bunch of inertial observers standing at the origin-event to travel 1 sec (according to each observer's wristwatch) with all possible velocities [in agreement with experiment], their endpoints (their "my watch reads 1 sec" events) trace out a hyperbola (the "circle" in Minkowski-spacetime).
$$(1\ \rm{sec})^2=(\Delta t)^2-(\Delta y/c)^2$$ [with my signature convention].

ghwellsjr
Gold Member
In a position-vs-time graph, if you ask a bunch of inertial observers standing at the origin-event to travel 1 sec (according to each observer's wristwatch) with all possible velocities [in agreement with experiment], their endpoints (their "my watch reads 1 sec" events) trace out a hyperbola (the "circle" in Minkowski-spacetime).
$$(1\ \rm{sec})^2=(\Delta t)^2-(\Delta y/c)^2$$ [with my signature convention].
In a recent thread, I posted a spacetime diagram that illustrates your point:

robphy
Homework Helper
Gold Member
In a recent thread, I posted a spacetime diagram that illustrates your point:
neat!

Please help me understand why there is a minus in the spacetime interval formula and not a plus.
It's not obvious from just a cursory glance. It is minus because that is what works in flat spacetime...much like all the formulas we choose to use...Although Einstein found that space and time can vary in different inertial frames, his math teacher Minkowski first replaced the [fixed] Euclidean distance with the [fixed ] spacetime interval....that is, recognized the negative sign!!! And so Einstein proceeded from there in developing GR.

In GR there is a different 'distance' measure, and in cosmology yet another.

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Not a hundered percent sure but I think the subtraction sign is there because it is opposite to the additions.

From a causality perspective length and time are equivalent, and opposite.

Add in a speed constant and it makes sense to calculate the interval between "happenings" treating time/length as equal but opposite.

For a physical occurance, either "its" closer (read length) to happening or "it" needs more time. Extend that to the idea of calculating a distance acoss a spacetime continuum with a speed constant.

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This is one of those threads where the response of the experts seems a little disappointing to me. The OP is hardly the first person to have struggled with this very point, and somewhere, I forget exactly where, but it is entirely possible that it was elsewhere on these forums, I have seen an altogether more satisfactory answer. For me, even the point that the Pythagoras Theorem, so familiar to most of us as something to do with two dimensional triangles, actually works across three dimensions – and indeed, theoretically only of course, across just however many dimensions you care to apply it – was itself a revelation. But when applying it to spatial dimensions, theoretical or actual, it is the sum of the squares of the individual dimensions you need to calculate. Why, with the time dimension does it suddenly become the difference? It seems a perfectly reasonable, and answerable question to me, with a perfectly solid basis in logic. Again, I recall someone pointing out that the first problem is that the time dimension uses different units than the spatial dimensions, and the presence of c in the time element is effectively as a conversion constant to normalise the time value to the space values. That seemed a very insightful point to me. And, as I said, I have seen an answer to the OP’s question that did seem to provide resolution at the time that I read it, though I confess that I cannot recover it in my mind at the moment. My strong feeling is that it relates to this point about the difference between the space and time dimensions and exactly how we measure them. I do get the point that sometimes, when somebody demands an answer in the manner of a baby demanding its pacifier, then it is not actually the most helpful to simply stick the dummy in their mouth. However, to dismiss this question as one of aesthetics does not seem entirely fair to me.

I am by no means an expert here, only an interested layman; my take on this is one of simple geometry. We know that the speed of light in vacuum is a function of permittivity and permeability of the vacuum; since these are fundamental constants, we would therefore expect that speed to be the same for all possible observers in space-time, regardless of their state of relative motion. In order for this to be possible, when we change relative speed between observers, we need to "trade" time for space, and vice versa, or else we couldn't maintain the ratio between measurements which give us the speed of light. If we were to plot time against space in a diagram, the speed of light ratio would therefore need to be an asymptote to our plot; the simplest function which fits this is a hyperbola, and the equation for a hyperbola has the general form

$$\displaystyle{x_{0}^{2}-...-x_{n}^{2}-c=0}$$

Hence the minus signs. Another way to look at it is that stationary observers should experience the longest proper time between given events A and B, since they experience no time dilation of any kind. Again, this is possible only if the time and space coefficients in the line element have opposite signs ( see twin paradox ).

However, to dismiss this question as one of aesthetics does not seem entirely fair to me.
Yea I found those two replies strange as well. That said aesthetics was in quotes so who knows what ZapperZ meant, surely it wasn't literal.

Oh and I found your reply disappointing as well. :tongue2: pervect's reply was really clear specifically "it's basically a restatement of the principle of the constancy of the speed of light." & WannabeNewton's mentioning to look into lightcones & the Minkowski metric are great replies to the OP.

the first problem is that the time dimension uses different units than the spatial dimensions, c in the time element is effectively a conversion constant to normalise the time value to the space values. That seemed a very insightful point to me.

The idea that time and space have different units and c is a "conversion" for natural units is insightful as to why there is a negative sign in the spacetime interval equation?

....it relates to this point about the difference between the space and time dimensions and exactly how we measure them.
I assure you it's not how we measure them but how they behave from a physics perspective, specifically as comparative measurements between "happenings"/distance across a spacetime continuum with a speed constant.

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I think the more interesting question is why the minus sign in the spacetime interval works so well. Obviously, we choose it because it matches observation well. But that's not a very satisfying answer.

My thoughts on the matter is that the minus sign gives us something very useful for modelling special relativity: In SR, there is a special speed, C, and so everything can be considered either slower than c, faster than c, or going at c. If you have a minus sign in the spacetime interval, you also have three categories: greater than 0, less than 0, or 0. However, if there was no minus sign, then the interval would always be greater than or equal to 0, and it would only be 0 if it was the interval between two identical points. So what is important here is that the minus sign introduces gives us a sufficiently complicated object that we can treat the speed of light as special.

For example, we say paths of length 0 are paths taken at the speed of light. Paths of length less than 0 are taken below the speed of light, and paths of length greater than 0 are taken faster than the speed of light. The next step is to realize that since the speed of light is constant in all reference frames, that changes of reference frame should be described by transformations that leave the spacetime interval constant (so that a path taken at the speed of light in one reference frame is also taken at the speed of light in another reference frame). This group of transformations is called SO(1,3), and is actually the set of lorentz transformations, which confirms that it is in fact a model of SR. By contrast, if we had all plus, then the group would have been SO(4) which isn't the set of lorentz transformations.

Edit: I don't like the idea that the spacetime interval is a restatement that the speed of light is constant. There are plenty of other spacetime intervals that make the speed of light constant, which are the various metrics of GR. Thus going from "speed of light is constant" to the minkowski metric is not trivial, and requires some extra assumptions (that it's a global symmetry).

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Another good explanation

Chestermiller
Mentor
The constancy of the speed of light implies that if the Lorentz interval is zero for one observer, it's zero for all. This works only because of the minus sign - it's basically a restatement of the principle of the constancy of the speed of light.

The stronger statement that the Lorentz interval is constant for all observers implies the above, for if it's constant and zero for one observer, it's zero for all. It turns out that the stronger formulation is true, but I'm not aware of any easy to explain reason why the stronger formulation turns out to be true.

Hopefully, this provides some insight into the motivation of why we need a minus sign, even if it doesn't totally explain it.
Here is a contrary view. In my judgement, for whatever it is worth, the constancy of the speed of light is not the cause of anything. It is one many the effects of the unique structural geometry of 4D space-time. Just because the development of special theory of relativity centered around the principle of relativity does not mean that it is the physical cause of anything. The geometry of 4D spacetime would be what it is even if there were no such thing as light.

Chet

I have not "explored" it but suspect c is a requirement of causality. And causality is a very simple logic.

In this specific context id call them synonymous.

there has been thread topics around this idea.

Oh and I found your reply disappointing as well. :tongue2: pervect's reply was really clear specifically "it's basically a restatement of the principle of the constancy of the speed of light." & WannabeNewton's mentioning to look into lightcones & the Minkowski metric are great replies to the OP.
Yes, I wasn’t commenting on the contributions of others, only on the contributions of those whose posts have a badge that carries with it certain expectations. I would again stress, I too am familiar with many posts by ZapperZ, and understand that his expertise is not just in this subject but as an educator. And any good educator knows that simply supplying answers is not a very effective way of educating. It is generally more effective to stimulate the questioner to think for themselves. I suppose his post may have been intended to do something of that kind, but I suppose I found myself stimulated to stand up for the OP in the face of what I found to be less than entirely valid criticism.

The idea that time and space have different units and c is a "conversion" for natural units is insightful as to why there is a negative sign in the spacetime interval equation?
No I didn’t mean insightful in that way. I suppose I was just highlighting some of the insights that had helped me to decode this particular formula, and suggesting that the insight that the OP needed was of a similar type and character, and doubtless one that both Vanadium 50 and ZapperZ would be perfectly capable of supplying, if they chose to.

I have not "explored" it but suspect c is a requirement of causality. And causality is a very simple logic.

In this specific context id call them synonymous.

there has been thread topics around this idea.
That's not quite true: There are spacetimes in general relativity that have closed timelike curves, despite the fact that the speed of light in GR is still constant. These curves essentially are the signature of time travel: you can meet up with an earlier version of yourself. Also consider that time travel is completely impossible in Newtonian mechanics, yet there is no maximum speed there (causality holds trivially in Newtonian mechanics).

I prefer to leave the constancy of the speed of light as an experimental/theoretical observation: either you have the michelson-morley experiment, or you have electrodynamics. Either (or both) of those can convince you to try to make a theory in which the speed of light is constant. I tried to give a good motivation for why you would consider a geometric theory in which the spacetime interval has a minus sign. Causal structure is built into the geometric structure of spacetime, so you still need to only consider the spacetime interval.

WannabeNewton
For anyone interested, I refer them to the book "General Relativity from A to B"-Robert Geroch. In particular, see the chapter titled "The Interval: The Fundamental Geometrical Object" where Geroch dedicates nearly 30 pages to about as good a derivation of the space-time interval of SR as you can probably get. This book was recommended to me by robphy a long while back.

Chestermiller
Mentor
That's not quite true: There are spacetimes in general relativity that have closed timelike curves, despite the fact that the speed of light in GR is still constant. These curves essentially are the signature of time travel: you can meet up with an earlier version of yourself. Also consider that time travel is completely impossible in Newtonian mechanics, yet there is no maximum speed there (causality holds trivially in Newtonian mechanics).

I prefer to leave the constancy of the speed of light as an experimental/theoretical observation: either you have the michelson-morley experiment, or you have electrodynamics. Either (or both) of those can convince you to try to make a theory in which the speed of light is constant. I tried to give a good motivation for why you would consider a geometric theory in which the spacetime interval has a minus sign. Causal structure is built into the geometric structure of spacetime, so you still need to only consider the spacetime interval.
This is precisely what I was trying to say, although you articulated it in a much better way than I did.

Chet