Why Is the Invariant Interval True for All Values of s?

[PreposterousUniverse]

If we use a light pulse that is emitted at A and absorbed at B. The spacetime interval between these two events are s'=s=0 in both frames of reference. But how does this invariance between s and s' extend to cases where s is not zero? Then we cannot measure the distance between the events using a light pulse and use the invariance of the speed of light to argue that s=s'. So how should we reason then? All explanations I have found just postulates that it is true in cases where s is some other value other than zero, but does not explain why it is true for all values of s.

 

[PeterDonis]

in both frames of reference

Which frames of reference? You haven't described any.

how does this invariance between s and s' extend to cases where s is not zero?

It works the same. The spacetime interval is always an invariant.

use the invariance of the speed of light to argue that s=s'.

We aren't using the invariance of the speed of light to argue that s = s'. The interval is invariant because it is an invariant; it's geometry.

All explanations I have found

Where have you looked?

 

[PreposterousUniverse]

Which frames of reference? You haven't described any.


It works the same. The spacetime interval is always an invariant.


We aren't using the invariance of the speed of light to argue that s = s'. The interval is invariant because it is an invariant; it's geometry.


Where have you looked?

Two inertial frames of reference. "The interval is invariant because it is invariant". What kind of explanation is that? Then why did we use the invariance of the speed of light to prove that s=s' in the case when s=0? Would it not also be true because an invariant is an invariant because of geometry then?

 

[Orodruin]

Why is Pythagoras’ theorem true?

The answer is the same: because that is how geometry of the underlying space works. The only difference is Euclidean vs Minkowski space.

 

[Ibix]

But how does this invariance between s and s' extend to cases where s is not zero?

When you ask how to get somewhere, the answer very much depends where you are starting from.

One obvious way to show the invariance of the interval is to write out the expression for $\Delta s^2$ and plug in the Lorentz transforms to eliminate the unprimed coordinates. But that might be a circular argument, depending on how you chose to derive the Lorentz transforms.

 

[PreposterousUniverse]

When you ask how to get somewhere, the answer very much depends where you are starting from.

One obvious way to show the invariance of the interval is to write out the expression for $\Delta s^2$ and plug in the Lorentz transforms to eliminate the unprimed coordinates. But that might be a circular argument, depending on how you chose to derive the Lorentz transforms.

Yes of course, that would be trivial. But circular reasoning as I see it, because I want to use the invariance of the interval to derive the Lorentz transformations. The case s=s'=0 for a light pulse traveling between point A and B follows from the principle that the speed of light is invariant in all inertial frames. But why does it extend to the case when s is not equal to zero?

 

[PreposterousUniverse]

Why is Pythagoras’ theorem true?

The answer is the same: because that is how geometry of the underlying space works. The only difference is Euclidean vs Minkowski space.

We haven't made any assumptions about the underlying geometry. We have only assumed that the speed of light is invariant. From this assumption we can prove that s=s'=0 for two event separated by a light pulse. But how does it extend to the case when s differs from 0?

 

[Orodruin]

We haven't made any assumptions about the underlying geometry. We have only assumed that the speed of light is invariant. From this assumption we can prove that s=s'=0 for two event separated by a light pulse. But how does it extend to the case when s differs from 0?

Who are ”we”? You need to be more specific about your assumptions if you are just going to randomly reject replies. The underlying space in SR is Minkowski space.

 

[PeterDonis]

Two inertial frames of reference.

Ok.

"The interval is invariant because it is invariant". What kind of explanation is that?

A geometric one.

Then why did we use the invariance of the speed of light to prove that s=s' in the case when s=0?

We didn't, as I have already told you. If you think "we" did, you need to give a specific reference that supports your claim.

Would it not also be true because an invariant is an invariant because of geometry then?

No "also" about it; it is true, for any interval, whether s = 0 or not, because of geometry.

 

[PeterDonis]

I want to use the invariance of the interval to derive the Lorentz transformations.

You can do that regardless of whether the interval is null. It is an easily demonstrated mathematical fact that the Lorentz transformations are the transformations that leave the Minkowski interval invariant in the general case, for any interval.

 

[PeroK]

If we use a light pulse that is emitted at A and absorbed at B. The spacetime interval between these two events are s'=s=0 in both frames of reference. But how does this invariance between s and s' extend to cases where s is not zero? Then we cannot measure the distance between the events using a light pulse and use the invariance of the speed of light to argue that s=s'. So how should we reason then? All explanations I have found just postulates that it is true in cases where s is some other value other than zero, but does not explain why it is true for all values of s.

This is covered in any SR textbook. The relevant chapter of Morin's book is free online:

https://scholar.harvard.edu/files/david-morin/files/cmchap11.pdf

This follows the traditional approach:

Postulates -> Lorentz Transformation -> Invariant Interval

 

[PreposterousUniverse]

Who are ”we”? You need to be more specific about your assumptions if you are just going to randomly reject replies. The underlying space in SR is Minkowski space.

I thought I was clear. We are not assuming the underlying space is Minkowski, we are going to derive it. We can derive from the relation s=s' which states that (cdt)^2-(dx)^2 = (cdt')^2 - (dx')^2. We can prove that this relation holds when (cdt)^2-(dx)^2 = (cdt')^2 - (dx')^2 = 0 by considering two events A and B separated by a light pulse and using the invariance of the speed of light. But we haven't proved that s=s' in general. We must show this before we can derive the general form of the transformations.

 

[PeterDonis]

We are not assuming the underlying space is Minkowski, we are going to derive it. We can derive from the relation s=s' which states that (cdt)^2-(dx)^2 = (cdt')^2 - (dx')^2.

But where did you get that relation from? From the Minkowski metric, i.e., from the fact that the underlying space is Minkowski.

We can prove that this relation holds when (cdt)^2-(dx)^2 = (cdt')^2 - (dx')^2 = 0 by considering two events A and B separated by a light pulse and using the invariance of the speed of light.

Which already assumes that the underlying space is Minkowski; otherwise the equation you are using for the interval is not correct.

Where are you getting all this from?

 

[PreposterousUniverse]

But where did you get that relation from? From the Minkowski metric, i.e., from the fact that the underlying space is Minkowski.


Which already assumes that the underlying space is Minkowski; otherwise the equation you are using for the interval is not correct.

Where are you getting all this from?

No we are not assuming a Minkowski metric or any other metric. (cdt)^2 - (dx)^2 is just a quantity that we are calculating. The space could be cartesian or any other space, we can calculate any quantity we want from the coordinates between two point in the space. But we have showed using the invariance of the speed of light that for two event connected by a light pulse, this particular quantity is invariant and equal to zero. But we haven't showed that it is invariant for other values. If we can do that, then we can derive the general transformations.

 

[PeterDonis]

we have showed using the invariance of the speed of light that for two event connected by a light pulse, this particular quantity is invariant and equal to zero

No you haven't, because you can't just assume that the coordinates of two events connected by a light pulse have the relationship you wrote down. You have to calculate that relationship from the metric--and unless the metric is the Minkowski metric, the relationship won't be the one you wrote down.

At this point I am closing this thread since you have given no references to back up any of your claims and what you are posting looks like personal speculation, which is off limits here. If you have an actual reference, you can PM me and I can review it to see if it's worth reopening the thread to discuss it.