# Spacetime interval and basic properties of light

• I
• HansH
Well, that's a long and complicated story. It is a consequence of the invariance of light speed, but it is not easy to see why. In summary, the space time interval is defined as ds^2=(cdt)^2-(dx^2+dy^2+dz^2), and it comes from the Lorentz Transformation and the invariance of the speed of light. This is a fundamental concept in relativity, and understanding its derivation can help deepen our understanding of the theory. Some sources start with the interval and derive other properties, while others start with the invariance of light speed. Either way, understanding why something is defined a certain way is important in fully grasping a concept.

#### HansH

TL;DR Summary
space time interval is defined as ds^2=(cdt)^2-(dx^2+dy^2+dz^2) but where does this equation come from?
While not having a professional physics background I was still interested in knowing more about special and general relativity. Therefore I was trying to find out where the space time interval was coming from in relation to the speed of light. Of course this is the first point to start I believe.

But in most explanations about special relativity it turned out that the spacetime interval was presented as something basic but without referring to where it exactly comes from. So for me that looked like making too much steps at a time and missing some fundamental steps in between especially where the minus sign comes from.
Therefore starting from scratch I tried to understand it starting from the speed of light being constant and the comparison between 2 reference frames with constant speed difference.
Then I could immediately see that pythagoras holds making the square of the distance (dx^2+dy^2+dz^2) + ds^2=(cdt)^2 where ds^2 is the distance that the light moves in 1 second as seen by each observer in its own reference frame. But the observer looking to the other reference frame sees the light passing over a longer distance therefore the light seems to go slower from his perspective.
so it this where it comes from?

• PeroK
space time interval is defined as ds^2=(cdt)^2-(dx^2+dy^2+dz^2) but where does this equation come from?

It is defined yes, but its motivation comes from the Lorentz Transformation, how time and spatial coordinates are measured in different inertial frames. If you have two events (coordinates in space-time) which are measured in two frames S and S'. Then ##\Delta s^2## has the same value in both frames. For a light ray, ##\Delta s^2 = 0##

So in order to understand why this is a sensible thing to calculate and why it is invariant, you might want to look for "derivation of the Lorentz transformation" and such.

It depends where you want to start.

Some sources like to start with the interval and derive various properties of Minkowski geometry, then simply say that experiment shows that this is an accurate model of the world. And you can show that Newtonian physics emerges as a low-speed approximation. This "geometry first" approach is one of the better ways to learn relativity, but it's very different from pre-relativistic physics.

Something nearer the historical path is to simply assert that the speed of light is invariant and insist on the principle of relativity. With a bit of work you can deduce the Lorentz transforms and show that the interval is invariant under Lorentz transforms.

So it's either a consequence of the invariance of light speed or light speed is a consequence of the invariance of the interval, depending on which assumptions you want to start with.

• cianfa72
No not specifically. Starting with something being defined in a certain ways always gave me the feeling that I missed some basic ideas behind because I think by starting with a definition its the wrong way around. you should first understand why something is defined as is. because there is always a good reason to define something and by starting with such definition seems to work back from something that you know and the one that you want to tell stil doesn't know 'so leaving him/her desparate. What I then always did was going back some steps and trying to find out the missing steps myself. But that of course takes unnecessary time because the original person that found out the theory for the first time also went through these steps in order to understand it.

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• Motore
Ibix said:
So it's either a consequence of the invariance of light speed or light speed is a consequence of the invariance of the interval, depending on which assumptions you want to start with.
How did Einstein himself start? I assume with the fact that the speed of light is constant for all observers. and that results in logical steps to come to the theory.

constant light speed as a consequence of the invariance of the interval to me sounds like reversing cause and result.

• PeroK
HansH said:
Summary: space time interval is defined as ds^2=(cdt)^2-(dx^2+dy^2+dz^2) but where does this equation come from?

missing some fundamental steps in between especially where the minus sign comes from.
That part is easy. If the speed of light is invariant then the formula for a light cone is invariant: $$-(c\Delta t)^2+ \Delta x^2+\Delta y^2 + \Delta z^2=0$$ in other words all frames agree that the same set of events defines a sphere expanding at ##c##. So that is where the minus sign comes from. It is immediately required for the second postulate.

From there it is a small jump to simply guess that maybe this holds for values other than ##0## on the right hand. Then we can derive all of the testable statements of SR and see if our guess is right.

$$-(c\Delta t)^2+ \Delta x^2+\Delta y^2 + \Delta z^2=0$$, a sphere expanding at ##c##. is there some format error here? looks rather cryptical to me.

HansH said:
How did Einstein himself start? I assume with the fact that the speed of light is constant for all observers. and that results in logical steps to come to the theory.
More or less. We understand the theory a lot better now, though, than Einstein did in 1905, and can fit it into the broader context of GR and other modern theories. So Einstein's way (which was supposed to convince experts in 19th century physics) isn't necessarily the best way to learn for a modern student.

HansH said:
$$-(c\Delta t)^2+ \Delta x^2+\Delta y^2 + \Delta z^2=0$$ is there some format error here?
No. It's just the observation that for a light ray, the distance traveled divided by the time taken is ##c##. Use Pythagoras and rearrange to get the formula stated. It must then be invariant if the speed of light is invariant.

HansH said:
$$-(c\Delta t)^2+ \Delta x^2+\Delta y^2 + \Delta z^2=0$$, a sphere expanding at ##c##. is there some format error here? looks rather cryptical to me.
No, this is the standard formula for a sphere of radius ##c\Delta t##. Haven’t you seen the formula for a sphere?

Ibix said:
More or less. We understand the theory a lot better now, though, than Einstein did in 1905, and can fit it into the broader context of GR and other modern theories. So Einstein's way (which was supposed to convince experts in 19th century physics) isn't necessarily the best way to learn for a modern student.
looking to myself during my study I always tried to understand things in logical steps and often found that a logical step was missing. then reconstructing that missing step almost always resulted in the conclusion that I gained a lost of insight but also lost a lot of time unnecessary and if it was explained in a slightly different way this could have been prevented. Trying to understand special and general relativity reminds me to that situation again.

Dale said:
No, this is the standard formula for a sphere of radius ##c\Delta t##. Haven’t you seen the formula for a sphere?
yes that is pythagoras x^2+y^2+z^2=r^2 but what does $$and ## mean and what is the relation between a sphere and the space time interval. I think i also here miss some thinking steps? HansH said: looking to myself during my study I always tried to understand things in logical steps and often found that a logical step was missing. then reconstructing that missing step almost always resulted in the conclusion that I gained a lost of insight but also lost a lot of time unnecessary and if it was explained in a slightly different way this could have been prevented. Trying to understand special and general relativity reminds me to that situation again. Well, your last thread was attempting to understand properties of curved spacetimes without knowing anything about flat spacetime. I'd say there were about two textbooks' worth of steps missing there. This topic, on the other hand, is simply about whether you choose to assume a Minkowski spacetime and deduce that the speed of light is invariant, or assume the speed of light is invariant and deduce a Minkowski spacetime. Neither approach is more or less logical, and the geometry-first approach is more in keeping with the way GR works. • PeroK, malawi_glenn and Vanadium 50 HansH said: what does$$ and ## mean
They mean that you need to refresh the page so that MathJax wakes up and renders the maths. Those symbols delimit LaTeX (see the guide linked below the reply box for details) but sometimes you need to refresh the page for it to render.

thanks, didn't know that. looks a lot better now.

Ibix said:
Well, your last thread was attempting to understand properties of curved spacetimes without knowing anything about flat spacetime.
I thought I was only talking about flat spacetime. At least that was where the question was about. Not sure where I used properties of curved spacetime?

Ibix said:
This topic, on the other hand, is simply about whether you choose to assume a Minkowski spacetime and deduce that the speed of light is invariant, or assume the speed of light is invariant and deduce a Minkowski spacetime.
This.

• malawi_glenn
HansH said:
what is the relation between a sphere and the space time interval
The first postulate gives you the invariance of the sphere with radius ##c\Delta t##, which is also known as the light cone. The light cone is a spacetime interval of 0.

HansH said:
How did Einstein himself start? I assume with the fact that the speed of light is constant for all observers. and that results in logical steps to come to the theory.
You have to start somewhere. Einstein started with his two postulates, and then derived the Lorentz transform. From there it is easy to prove the invariance of the spacetime interval.

I actually prefer to start with the invariance of the spacetime interval. Then it is possible but not easy to derive the Lorentz transform and the two postulates, but more importantly you can use that to easily go to a geometrical treatment of spacetime and a tensor-based treatment of physics.

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HansH said:
How did Einstein himself start? I assume with the fact that the speed of light is constant for all observers. and that results in logical steps to come to the theory.
He started from the "Relativity Principle" (RP): the laws of physics are identical in all inertial (nonaccelerating) frames, and the "Light Principle" (LP): Light signals in vacuum are propagated rectilinearly, with the same speed ##c## at all times, in all directions, in all inertial frames.

HansH said:
constant light speed as a consequence of the invariance of the interval to me sounds like reversing cause and result.
That's my opinion too -- I prefer to start from the (imho more intuitive) RP.

But it turns out that the LP is unnecessary. There are many threads here on PF that discuss this (so-called) "1-postulate derivation" of Special Relativity. I'm not sure what your math level is, but since you posted this as an I-level thread, here's some links to older posts/threads where I've discussed this before.

https://www.physicsforums.com/threa...-1-spacetime-only.1000831/page-2#post-6468652
Post #44.

Post #6.

Post #26.

• • pinball1970 and PeroK
I like Bondi's presentation of relativity.
The square-interval comes out from physically motivated "radar measurements".
It turns out that the square-interval between two timelike-related events is equal to
the area of a particular parallelogram (a "causal diamond") on a spacetime diagram,
where the two events are at the corners of the timelike-diagonal.

(The presentation of the square-interval resembling the Pythagorean theorem with a sign difference between space and time arises as a corollary of Bondi's presentation [in the eigenbasis: light-cone coordinates].)

See

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• Dale
This.
Not sure where I used

HansH said:
looking to myself during my study I always tried to understand things in logical steps and often found that a logical step was missing. then reconstructing that missing step almost always resulted in the conclusion that I gained a lost of insight but also lost a lot of time unnecessary and if it was explained in a slightly different way this could have been prevented. Trying to understand special and general relativity reminds me to that situation again.
Physics in the end is about describing how things work. I can propose any wild theory I want by definition. That will however most likely not be a goid description of observation and therefore ruled out and discarded by experiments. The trick is to find a definition that is a good description. It is not necessarily the case that you can purely reason from logic to arrive at that description. You can however try based on some experimental result and assumptions and it will sometimes work and other times not. The main thing to understand though is why a theory, defined as it is, gives a particular set of predictions.

• Dale and PeroK
Orodruin said:
Physics in the end is about describing how things work. I can propose any wild theory I want by definition. That will however most likely not be a goid description of observation and therefore ruled out and discarded by experiments. The trick is to find a definition that is a good description. It is not necessarily the case that you can purely reason from logic to arrive at that description. You can however try based on some experimental result and assumptions and it will sometimes work and other times not. The main thing to understand though is why a theory, defined as it is, gives a particular set of predictions.
This is exactly what I mean. so you cannot start with a definition but should start with how things work and then the definition is the result of that. For me that is the only way to learn things because starting an explanation with a definition directly gives me the feeling that I mis something important underlying. but pratice of teaching is ofthen not that way unfortunately.

based on 'start with how things work ' I come to the logical sequence of understanding as follows: HansH said:
based on 'start with how things work ' I come to the logical sequence of understanding as follows:
Your line segments are labeled wrong in your "moving frame" diagrams. ##ds## is the invariant length of the line segment that is vertical in your first frame and the hypotenuse of the triangle in your moving frames. The vertical lines you have labeled as ##ds## in your moving frame diagrams should be labeled ##cdt'## and ##cdt''##.

PeterDonis said:
you mean this? but then it would mean that according to pythagoras ds^2=dx^2+ct^2 which is a different equation. so then I am confused. HansH said:
This is exactly what I mean. so you cannot start with a definition but should start with how things work and then the definition is the result of that.
No, then you are misunderstanding my post. You can perfectly well start with a definition. It is not at all the case that you will necessarily be able to reason logically based on whatever results you have available.
HansH said:
For me that is the only way to learn things because starting an explanation with a definition directly gives me the feeling that I mis something important underlying. but pratice of teaching is ofthen not that way unfortunately.
The important thing is not really how you arrive at a theory. It is what the theory predicts and understanding why it makes those predictions that is of importance.

• Dale
HansH said:
No not specifically. Starting with something being defined in a certain ways always gave me the feeling that I missed some basic ideas behind because I think by starting with a definition its the wrong way around. you should first understand why something is defined as is. because there is always a good reason to define something and by starting with such definition seems to work back from something that you know and the one that you want to tell stil doesn't know 'so leaving him/her desparate. What I then always did was going back some steps and trying to find out the missing steps myself. But that of course takes unnecessary time because the original person that found out the theory for the first time also went through these steps in order to understand it.
If you consult any decent textbook on SR, all this will be explained. I like Helliwell's book:

There is also Morin, the first chapter of which is available here:

https://scholar.harvard.edu/files/david-morin/files/relativity_chap_1.pdf

HansH said:
you mean this? but then it would mean that according to pythagoras ds^2=dx^2+ct^2 which is a different equation. so then I am confused. View attachment 305642
Pythagoras’ theorem is a theorem in Euclidean space. Spacetime is not Euclidean, it is Minkowski space. In Minkowski space, the equivalent of Pythagoras’ theorem is ##ds^2 = c^2 dt^2 - dx^2##.

Orodruin said:
Pythagoras’ theorem is a theorem in Euclidean space. Spacetime is not Euclidean, it is Minkowski space. In Minkowski space, the equivalent of Pythagoras’ theorem is ##ds^2 = c^2 dt^2 - dx^2##.
ok, but if you say :In Minkowski space, the equivalent of Pythagoras’ theorem is ##ds^2 = c^2 dt^2 - dx^2## : then I am back to the openings question of the topic: why, because I still do not understand. so it seems difficult to get the basic idea clearly explained at headlines without diving into the books while I hoped this is what the forum could add. I thought I understood but seems to be on the wrong track, so I think I will first check the links in #29

HansH said:
this link does not seem to be accessible without giving away personal details to third parties.
My suggestion was that you buy the book!

• pinball1970
HansH said:
This is exactly what I mean. so you cannot start with a definition but should start with how things work and then the definition is the result of that. For me that is the only way to learn things because starting an explanation with a definition directly gives me the feeling that I mis something important underlying. but pratice of teaching is ofthen not that way unfortunately.
The trouble with that in this case is that Newton's simple theories work just fine in almost everything back then in 1905. Einstein came up with his theory first and experimental results came later. The theory has no grip in everyday life. To this day the world of woe harbors skeptics numerous.

I find Einstein's original paper very down to earth, you might like it. He said he was largely motivated by Fizeau's 1851 experiment on the speed of light in flowing water and by the idea that Maxwell's equations were valid no matter what.

HansH said:
then I am back to the openings question of the topic: why, because I still do not understand.
The question you should be asking is why this gives a good description of nature and makes the speed of light invariant. The answer was given by @Dale in post #7.