# Invariant interval

• I
In SR why is the following length-interval invariant

(1) $$0 = (cdt)^{2}-dx^{2}$$

While,

(2) $$0 = (cdt)^{2}+dx^{2}$$

is not invariant?

The first expressions (1) measures the coordinates of a wavefront propagating away from the observer with the speed och light, and since c is a constant, this expression must be zero in all reference frames moving with constant velocity relative to it.

But why not use expression (2) as a way to measure distans?

Because experiments show that it won't work.

Nugatory
Mentor
In SR why is the following length-interval invariant
(1) ## 0 = (cdt)^{2}-dx^{2}##
While,
(2) ## 0 = (cdt)^{2}+dx^{2} ##
is not invariant?
Try applying the Lorentz transformations to ##(cdt)^2-(dx)^2## and to ##(cdt)^2+(dx)^2##. The first one remains the same when you transform into the primed coordinates while the second one does not; therefore the first one is invariant and the second is not.

The Galilean transforms work the other way: the second and not the first is invariant under the Galilean transforms. Thus, your question
why not use as expression (2) as a way to measure distance?
comes down to asking whether the Galilean or the Lorentz transformations work properly with the geometry of spacetime; experiment shows that the Lorentz transforms are the ones that work.

Orodruin
Staff Emeritus
Homework Helper
Gold Member
The Galilean transforms work the other way: the second and not the first is invariant under the Galilean transforms.
Neither is invariant under Galilei transformations. The basics behind Galilei transforms stipulate that ##dt## is invariant (absolute time) and that ##d\vec x^2## is invariant if ##dt = 0##.

Nugatory
Mentor
Neither is invariant under Galilei transformations. The basics behind Galilei transforms stipulate that ##dt## is invariant (absolute time) and that ##d\vec x^2## is invariant if ##dt = 0##.
Ah - yes, of course.

While,

(2) $$0 = (cdt)^{2}+dx^{2}$$

is not invariant?

Because your definition yields: ##dt=dx=0##, hence ##c=\frac{0} {0}## which is undetermined.

Basically, what you are interested in the above model are rotations of the Euclidean plane, and you get the corresponding transformations and indeed there is no constant speed of light there under those transformations to be defined.

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(1)
0 = (cdt)^{2}-dx^{2}

means if $$\frac{dx}{dt}=c$$ in one frame of reference then $$\frac{dx'}{dt'}=c$$ stands in any frame of reference. (1) is meaning invariance of maximum propagation speed or light speed. (2) does not hold.

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haushofer
Neither is invariant under Galilei transformations. The basics behind Galilei transforms stipulate that ##dt## is invariant (absolute time) and that ##d\vec x^2## is invariant if ##dt = 0##.
This is the reason why in Newton Cartan theory one needs degenerate metric structures, giving subtleties wrt Einstein's theory.

haushofer
About the question: in SR one has an invariant speed c. The minus sign is because we define the variations of space and time as positive. The two intervals you mention are connected by an analytic extension of time in the complex plane.

About the question: in SR one has an invariant speed c. The minus sign is because we define the variations of space and time as positive. The two intervals you mention are connected by an analytic extension of time in the complex plane.

So could I derive SR by the requirement that the expression

$$(cdt)^{2}+(dx)^{2}$$

schould be invariant. That is if
$$(cdt)^{2}+(dx)^{2} = (cdt')^{2}+(dx')^{2}$$

About the question: in SR one has an invariant speed c. The minus sign is because we define the variations of space and time as positive. The two intervals you mention are connected by an analytic extension of time in the complex plane.
The same. If you regard time or space as imaginary, you fall back to the (1) case.

So could I derive SR by the requirement that the expression

$$(cdt)^{2}+(dx)^{2}$$

schould be invariant. That is if
$$(cdt)^{2}+(dx)^{2} = (cdt')^{2}+(dx')^{2}$$

Both distances (1) and (2) are invariant under Lorentz transformations to be defined as Hyperbolic rotations of the Minkowski plane in the (1) case and ordinary rotations of the Euclidean plane in the (2) case.

As I've already explained to you in #6 post the (1) case yields invariant speed of light, the (2) case yields nonsense.

PeterDonis
Mentor
2020 Award
could I derive SR by the requirement that the expression
$$(cdt)^{2}+(dx)^{2}$$
schould be invariant.

No. The theory you would get in this case is ordinary Euclidean geometry (with funny units for one coordinate), not SR.

haushofer
Well, technically you probably could do it, and perform an analytic extension to go from Euclidean to Minkowksi spacetime and from so(4) to so(3,1) afterwards, but that's beyond the level of this topic I guess :P

And cumbersome.

Mister T
Gold Member
In SR why is the following length-interval invariant

(1) $$0 = (cdt)^{2}-dx^{2}$$

Because ##c## is an invariant speed.

While,

(2) $$0 = (cdt)^{2}+dx^{2}$$

is not invariant?

Because ##c## is an invariant speed.

But why not use expression (2) as a way to measure distans?

You would use a clock to measure the time, and then multiply the result of that measurement by ##c##. So you need an expression consistent with that process. Expression (1) is, but expression (2) isn't.

pervect
Staff Emeritus
In SR why is the following length-interval invariant

The first expressions (1) measures the coordinates of a wavefront propagating away from the observer with the speed och light, and since c is a constant, this expression must be zero in all reference frames moving with constant velocity relative to it.

But why not use expression (2) as a way to measure distans?

Why would that make any sense?

I see how one could assume 1) or 2), but not how you could possibly assume both. You've already pointed out though, that only one relationship, the first, can explain the fact that the speed of light is constant in all frames, something that we observe.

I can imagine working out the consequences of 2) on how the speed of light must transform between frames. I can't imagine what your proposal of combining 1) and 2) would even mean, unfortunately. I would suggest, if you're really motivated, try to derive something physical, like the frame-dependent speed of light, or the velocity addition law, from your assumptions. If this seems too puzzling to carry out, it's probably a sign that you need to study relativity more to see how it would do the same job, then apply the techniques from relativity to your theory.

The relativistic techniques would basically involve noting that the appropriate coordinate transforms for case 1 were the Lorentz transform, which, by setting c=1, can be written as

$$x' = \cosh \theta \, x - \sinh \theta \,t \quad t' = \cosh \theta \, t - \sinh \theta \, x$$

Here we regard ##\theta## as a parameter, it has a name, 'rapidity'. See for instance https://en.wikipedia.org/wiki/Rapidity.
The transforms for case 2 can be written as

$$x' = \cos \theta \, x - \sin \theta \,t \quad t' = \cos \theta \, t - \sin \theta \, x$$

We might recognize it as a rotation, and the formula as saying that rotation doesn't change distances.

We can compute that ##x'^2 - t'^2 = x^2 -t^2## in case 1, and that ##x'2 + t'^2 = x^2 + t^2## in case 2, and note that the transformation laws make them both invariant.

So rotations preserve Euclidean distances in case 2, the hyperbolic 'rotations' of the Loretnz boost preserve the Lorentz interval in case 1. But I don't see at all how you are proposing to mix the cases together, you'd need to write up something more mathematical before it would even make sense to me.