I Spacetime interval and basic properties of light

[HansH]

Let's say i stand in the origin and shoot a ray of light in the +x direction. After t = 1s we have that x=ct, after t = 2s then x=2ct, and so on. Therefore (ct)² = x² which is equivalent to 0 = (ct)² - x².
Is this what you are trying to understand? Why there is a minus sign in front of x²?

yes I think so. I assume your description with one dimension where the light goes to is same as what Dale showed me with 3 dimensions giving a sphere of light at distance ct. But how should I interpret that in general, so when ds is not 0, especially in relation to both observers.

 

[malawi_glenn]

yes I think so. I assume your description with one dimension where the light goes to is same as what Dale showed me with 3 dimensions giving a sphere of light at distance ct. But how should I interpret that in general, so when ds is not 0, especially in relation to both observers.

Yes it is, the 3D-case follows from that (spherical wavefront)

For other pair of events which do not have "light like" separation, you consult post #37
It follows from the Lorentz transformation that $s^2 = (ct)^2 - x^2$ (1) is invariant (this is the "1D" case since we start with just a boost in the x-direction here).
So just by imposing same speed of light in all inertial frames and the linear transformation, the lorentz-transformation and this result (1) pops out.

Here is an example.
Two flashes of lightning is measured to occur simultanously in frame $S$. They are separated with 250 m along the x-axis. A spaceship is traveling at 0.60c relative $S$ in the x-direction.
In the $S$ frame we have these coordinates $t_1 = t_2$ and $x_2 = x_1+250$ m.
In $S$ the space-time interval (squared) for these two events is $c(t_1-t_2)^2 - (x_1-x_2)^2 = 250^2$ m²
What are the time- and spatial coordinates for these two events in the spaceship frame $\tilde S$?
We consult the Lorentz-tranformation. The gamma factor is 1.25 (I am pretty sure you can calculate it yourself).
$\tilde x_1 = \gamma (x_1 - vt_1) \:,\: \tilde x_2 = \gamma (x_2 - vt_2) \:,\: \tilde t_1 = \gamma (t_1 - x_1v/c^2)\:,\:\tilde t_2 = \gamma (t_2 - x_2v/c^2)$
We can now calculate the space-time interval (squared) as measured in frame $\tilde S$.
Since we already know that this is the same as we had in $S$ we do not need to perform any calculations, we have that $c(\tilde t_1-\tilde t_2)^2 - (\tilde x_1-\tilde x_2)^2 = 250^2$ m² also!
I leave that as an excerise for you to actually compute this :)

Note that we have not mentioned observers yet, just coordinates measured in the different frames. For a stationary observer you also need to account for the time it takes for the information to travel to him/her.

 

[Dale]

so not the ''null one''

OK, so if I am hearing you right, you now understand where the minus sign for the $-c^2\Delta t^2$ comes from in my steps 1)-3), and now you want to move on to point 4). Is that correct?

 

[HansH]

For other pair of events which do not have "light like" separation, you consult post #37
It follows from the Lorentz transformation that $s^2 = (ct)^2 - x^2$ (1) is invariant (this is the "1D" case since we start with just a boost in the x-direction here).

now you introduce a term "light like" separation, which is new to me. The opposite is space like I assume? (at least both terms I recognize in general as going slow or very fast) probably better if I first do a more thorough reading of special relativity, because I see I keep your hole team busy which is probably too much asked and this for me problematic point about the minus sign now seems to be almost clear and after al not difficult. the most difficult part was probably not knowing what was the thought behind. now I think i know. simply an expanding lightcone and the relation between time and position. how simple can it be. small time gives still small expansion of the lightcone, large time gives large expansion so the difference scaled with the proper factor (c) is constant. I will continue with checking #37

 

[HansH]

OK, so if I am hearing you right, you now understand where the minus sign for the $-c^2\Delta t^2$ comes from in my steps 1)-3), and now you want to move on to point 4). Is that correct?

yes, especially what exactly ds means and its consequences for reference frames and lightcones I have the feeling that I stil mis something there but not easy to describe what.

 

[malawi_glenn]

now you introduce a term "light like" separation, which is new to me. The opposite is space like I assume?

Yeah "light-like" means that $\Delta s = 0$. I forgot to add "and time-like" separations.
You have three cases for what $\Delta s^2$ can compute to: zero, greater than zero, smaller than zero.

• Space-like separation means that the spatial separation between two events is greater than that of the distance a light ray would travel, i.e. $(\Delta x)^2> (c\Delta t)^2$ (outside the light-cone)
• Time-like separation means that the spatial separation between two events is smaller than that of the distance a light ray would travel, i.e. $(\Delta x)^2 < (c\Delta t)^2$ (inside the light-cone)
• Light-like separation means that $(\Delta x)^2 = (c\Delta t)^2$ (on the light-cone)

probably better if I first do a more thorough reading of special relativity

Just remember that there are two ways one can start.
1) define the space-time interval to be invariant, then equal speed of light in all frames pops out.
2) define equal speed of light in all frames pops out, then invariant space-time interval pops out.
Both ways leads to equivalent physics.

 

[Dale]

so not the ''null one'' so for my understanding this means that at t=0 the light already is at a radius ds^2

So light is only for the null interval. Any non-null interval refers to something other than light.

The key thing to understand is that $\Delta s^2$ is a sort of “distance”. It is a ”distance” that includes actual standard spatial distances measured by rulers ($\Delta s^2>0$) which are called spacelike, and it includes temporal “distances” measured by clocks ($\Delta s^2<0$) which are called timelike.

So if $\Delta s^2>0$ at $\Delta t=0$ you have a sphere which consists of all of the points that are a ruler-measured distance of $\Delta s$ away from the initial point.

Or if $\Delta s^2<0$ at $\Delta x=\Delta y=\Delta z=0$ then you have the two events that are a clock-measured “distance” of $\Delta \tau =\sqrt{-\Delta s^2/c^2}$ away from the initial event.

the question is then: where are both observers at t=0? are they still both in the same point or not?

The observers are not important here. One of the benefits of this approach is that you can focus on the physics of what is actually physically happening, and focus less on who sees what.

It doesn’t matter where the observers are located or what reference frame they are using, they will agree on $\Delta s^2$. Perhaps a concrete example will help.

Suppose in some reference frame that we have two firecrackers, one at $x=0$ and the other at $x=10$, that explode simultaneously at $t=0$. So $\Delta x=10$ and $\Delta t=0$ and plugging that into the spacetime interval formula gives $\Delta s^2=100$.

Now, in a reference frame moving at $v=0.6c$ we can use the Lorentz transform to get $\Delta x’=12.5$ and $\Delta t’= 7.5$ and plugging that into the spacetime interval formula gives $\Delta s^2=100$. So even though the two frames disagree on the time between the two firecrackers exploding and even though they disagree on the distance between the explosions, they agree on the spacetime interval between the explosions.

 

[PeterDonis]

now you introduce a term "light like" separation, which is new to me.

@malawi_glenn gives a good explanation in post #96. The one point I would add is that this threefold classification of intervals into timelike, lightlike, and spacelike has no counterpart in Euclidean geometry or Newtonian mechanics. In Euclidean geometry $ds^2$ is always positive. So the whole timelike, lightlike, spacelike thing is unlike anything you are used to and you should be very wary of trying to make analogies about it with anything you already know.

 

[HansH]

For other pair of events which do not have "light like" separation, you consult post #37
It follows from the Lorentz transformation that $s^2 = (ct)^2 - x^2$ (1) is invariant (this is the "1D" case since we start with just a boost in the x-direction here).

now you introduce a term "light like" separation, which is new to me. The opposite is space like I assume? (at least both terms I recognize in general as going slow or very fast) probably better if I first do a more thorough reading of special relativity, because I see I keep your hole team busy which is probably too much asked and this for me problematic point about the minus sign now seems to be almost clear and after al not difficult. the most difficult part was probably not knowing what was the thought behind. now I think i know. simply an expanding lightcone and the relation between time and position. how simple can it be. small time gives still small expansion of the lightcone, large time gives large expansion so the difference scaled with the proper factor (c) is constant. I will check #37

So light is only for the null interval. Any non-null interval refers to something other than light.

Perhaps a concrete example will help.

thanks, this really helps.

 

[HansH]

Here are some good notes by Shankar from Yale Open Courses https://oyc.yale.edu/sites/default/files/relativity_notes_2006_5.pdf
video: https://oyc.yale.edu/physics/phys-200/lecture-12Consider .....
Conserved /

Now there are plenty of other invariant quantities in special relativity (such as the invariant mass) which can be proven to be so following a smilar calculation as above. But, it is much nicer to work with four-vector formalism.

good explanation of the concept of invariance. This helps. It looks like magic that ds is invariant. I have the feeling however that the invariance of ds is no coincidence and should follow in a logical way from the general thoughts behind the whole idea.

 

[HansH]

@malawi_glenn gives a good explanation in post #96. The one point I would add is that this threefold classification of intervals into timelike, lightlike, and spacelike has no counterpart in Euclidean geometry or Newtonian mechanics. In Euclidean geometry $ds^2$ is always positive. So the whole timelike, lightlike, spacelike thing is unlike anything you are used to and you should be very wary of trying to make analogies about it with anything you already know.

you say Euclidean geometry, but if I understand that well it means not curved. but special relativity is also not curved. and this topic is about special relativity. So do I mis something here? or do you mean 4D flat spacetime not being part of Euclidean geometry?

 

[PeterDonis]

you say Euclidean geometry, but if I understand that well it means not curved.

"Not curved" is one of the properties of Euclidean geometry, yes, but it's hardly the only one. Another one is "positive definite metric", which means what I said, that $ds^2$ is always positive.

special relativity is also not curved

That's correct, Minkowski spacetime is also not curved. It happens to share that property with Euclidean geometry. But it does not share other properties with Euclidean geometry; in particular, $ds^2$ is not always positive in Minkowski spacetime.

 

[robphy]

@malawi_glenn gives a good explanation in post #96. The one point I would add is that this threefold classification of intervals into timelike, lightlike, and spacelike has no counterpart in Euclidean geometry or Newtonian mechanics.

In general, starting with "[future]-timelike" (along inertial worldlines [which are arguable more physical than spatial lines])
I define "spacelike" to be orthogonal to "timelike"
and "null" to be eigenvectors of the boost.
(Using the metric (a figure representing the unit "circle": circle, "$\tau=1$" hyperbola, or "t=1-hyperplane"),
the spacelike direction is the tangent to the "circle" where the timelike radius vector meets the "circle".)With this scheme, the classification still holds for the Newtonian/Galilean case, with "spacelike" and "null" coinciding (as seen in the "opening up the light-cone" construction of the Newtonian/Galilean limit).
This seems consistent with the Cayley-Klein classification of geometries,
as seen in the diagram in my answer to
https://physics.stackexchange.com/q...ids-5-postulates-false-in-minkowski-spacetime
that was linked in #83 above.
(Yaglom 's "ordinary lines" or "lines of the first kind" are "timelike",
those of the "second kind" are "spacelike", and
the "special lines" are "null" (or "lightlike" in special relativity).
Ref: p. 184 of A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity
https://www.google.com/search?q="lines+of+the+second+kind"+yaglom ).

 

[PeterDonis]

I define "spacelike" to be orthogonal to "timelike"

This is much too restrictive for MInkowski spacetime. Most spacelike vectors are not orthogonal to most timelike vectors.

and "null" to be eigenvectors of the boost.

This part is fine, and is in fact a good way of looking at how the action of boosts is fundamentally different for null vectors as compared to spacelike and timelike vectors; boosts rotate (hyperbolically) the latter but dilate the former.

With this scheme, the classification still holds for the Newtonian/Galilean case, with "spacelike" and "null" coinciding

I'm not sure how this works for the Newtonian/Galilean case, since AFAIK there is not a single vector space that includes all of the relevant vectors and has a meaningful definition of orthogonality.

 

[robphy]

This is much too restrictive for MInkowski spacetime. Most spacelike vectors are not orthogonal to most timelike vectors.

If a vector is spacelike (which is an invariant notion), there is a timelike vector orthogonal to it.
All one needs is one timelike vector.

For clarity, maybe I should rephrase:​

a "spacelike" vector is defined as a vector that is orthogonal to some timelike vector.​

(Geometrically, fix a point-event...​

a "timelike" vector from that point-event is a radius vector to a "circle" centered at that point-event ;​

a "spacelike" vector from that point-event is a vector parallel to the tangent to a "circle" centered at that point-event)​

I'm not sure how this works for the Newtonian/Galilean case, since AFAIK there is not even a single vector space that includes all of the relevant vectors.

I'm not sure what you mean here.
A position-vs-time graph (with its Newtonian/Galilean structure), a Minkowski spacetime diagram, and the usual Euclidean plane
all have the structure of an affine space.

 

[Dale]

It looks like magic that ds is invariant. I have the feeling however that the invariance of ds is no coincidence and should follow in a logical way from the general thoughts behind the whole idea.

If you start with the two postulates of relativity, then you can derive the Lorentz transform. Once you have the Lorentz transform you can easily prove the invariance of the spacetime interval.

I just prefer to take a different approach: Just start with the spacetime interval and examine the consequences. You have to start with something, so I like starting with the one formula that you can do the most with. Because you started with one thing, if you find anything that doesn’t match experiments then you know exactly what went wrong, there is only one place to look.

 

[robphy]

good explanation of the concept of invariance. This helps. It looks like magic that ds is invariant. I have the feeling however that the invariance of ds is no coincidence and should follow in a logical way from the general thoughts behind the whole idea.

Here's how I think of the structure (as inspired by the various relativity courses I took and the books I consult).

Euclidean geometry, position-vs-time graphs (with the Newtonian/Galilean structure), and the Minkowski spacetime diagram all have the structure of an affine space (for simplicity, make it a "vector space").

With a vector space, we can add and scalar-multiply.
We can compare the relative sizes of parallel vectors.
But we can't assign a length or magnitude to a vector.

We need additional structure: a metric (a "circle"), some kind of structure [a tensor] that assigns a square-magnitude to a vector.
We can represent this by a https://en.wikipedia.org/wiki/Quadric .
Choose one to represent the tips of all vectors from the origin to be declared as "unit vectors".
The equation of this figure effectively chooses the form of the "invariant" quantity.

(This is similar to the notion of doing
https://en.wikipedia.org/wiki/Straightedge_and_compass_construction
or https://en.wikipedia.org/wiki/Poncelet–Steiner_theorem
rather than straightedge-only constructions.)

(The metric "circle" also defines when two vectors are orthogonal,
as well as provide a measure of angles between radius vectors,
and maybe angles between vectors tangent to the "circles").

If you ask for transformations that preserve that quadric (i.e. keep the magnitudes of vectors unchanged),
you have determined the "boosts" or "rotations".
(related: https://en.wikipedia.org/wiki/Erlangen_program
https://encyclopediaofmath.org/wiki/Erlangen_program )

Example: Modify the E-slider to choose the metric figure
https://www.desmos.com/calculator/kv8szi3ic8

For special relativity, rather than choose a distinguished hyperbola,
I think I can choose a distinguished parallelogram (what will become a "light-clock diamond").

 

[PeterDonis]

All one needs is one timelike vector.

Why? If you're going to consider boosts, you have to have multiple timelike vectors since boosts map timelike vectors to different timelike vectors.

A position-vs-time graph (with its Newtonian/Galilean structure), a Minkowski spacetime diagram, and the usual Euclidean plane
all have the structure of an affine space.

Affine space structure is not enough, is it? Boosts are transformations on a vector space, not on an affine space.

 

[robphy]

All one needs is one timelike vector.

Why? If you're going to consider boosts, you have to have multiple timelike vectors since boosts map timelike vectors to different timelike vectors.

Maybe my phrasing is incomplete.
I'll repeat my revision from #105:

For clarity, maybe I should rephrase:​

a "spacelike" vector is defined as a vector that is orthogonal to some timelike vector.​

(Geometrically, fix a point-event...​

a "timelike" vector from that point-event is a radius vector to a "circle" centered at that point-event ;​

a "spacelike" vector from that point-event is a vector parallel to the tangent to a "circle" centered at that point-event)​

​

In the revised definition,
the spacelike vectors from the origin (which are parallel to tangent-vectors to the metric "circle" centered at the origin)

• in special relativity, point outside of the light-cone (and can regarded as radius vectors to another hyperboloid)
• in Newtonian/Galilean spacetime, point tangent to the absolute-simultaneity hyperplane

as seen in https://www.desmos.com/calculator/kv8szi3ic8 (vary the E-slider).

 

[PeterDonis]

a "spacelike" vector is defined as a vector that is orthogonal to some timelike vector.

Yes, this part I understand.

(Geometrically, fix a point-event...a "timelike" vector from that point-event is a radius vector to a "circle" centered at that point-event ;a "spacelike" vector from that point-event is a vector parallel to the tangent to a "circle" centered at that point-event)

This part I don't understand.

 

[robphy]

(Geometrically, fix a point-event...​

a "timelike" vector from that point-event is a radius vector to a "circle" centered at that point-event ;​

a "spacelike" vector from that point-event is a vector parallel to the tangent to a "circle" centered at that point-event)​

This part I don't understand.

It's the idea that orthogonality is defined by generalizing the idea
that: at a point of a circle, the tangent is orthogonal to the radius.

So, once we have determined a vector is spatial according to some inertial observer,
then it is spacelike (it points outside of the light cone).

(From https://www.physicsforums.com/threads/minkowski-normal-v-euclidean-normal.742230/post-4683143 ...)

As Minkowski suggests (
https://en.wikisource.org/wiki/Translation:Space_and_Time (near [9] )
https://en.wikisource.org/wiki/Page:De_Raum_Zeit_Minkowski_018.jpg ),
where he uses "normal" instead of "orthogonal" or "perpendicular"

Let us decompose a vector drawn from O towards x, y, z, t into its four components x, y, z, t. If the directions of the two vectors are respectively the directions of the radius vector OR of O at one of the surfaces $\pm F=1,$ and additionally a tangent RS at the point R of the relevant surface, then the vectors shall be called normal to each other. Accordingly
$$c^{2}tt_{1}-xx_{1}-yy_{1}-zz_{1}=0,$$ which is the condition that the vectors with the components x, y, z, t and $\displaystyle \left(x_{1}\ y_{1}\ z_{1}\ t_{1}\right)$ are normal to each other.

This construction extends to the Newtonian/Galilean case where their "spacelike vectors" are tangent to its "Galilean circle". In this case, the "Galilean unit circle" and "t=1" hyperplane of simultaneity are on the same hyperplane. (The eigenvectors of a Galilean boost (the Galilean-null vectors) also lie on this hyperplane.)

 

[PeterDonis]

Yes, this part I understand.This part I don't understand.

Perhaps it would help if I described my own understanding of how the Galilean case works. Galilean "spacetime" is a stack of Euclidean 3-spaces, each labeled with a real number $t$ that is its "absolute time". There are two classes of vectors in this spacetime: vectors which are tangent to one of the Euclidean 3-spaces, and vectors which are not. One could label the first class as "spacelike" and the second as "timelike", but to me this labeling is somewhat misleading since there is no spacetime metric for Galilean spacetime so there is no correspondence with the classification of vectors by the sign of their squared norm that we have in Minkowski spacetime.

As for orthogonality, since there is no spacetime metric, if we are going to define orthogonality at all, the only way to do it would seem to be to say that every spacelike vector is orthogonal to every timelike vector. It is also possible, of course, to have spacelike vectors that are orthogonal to other spacelike vectors. This means that the statement that every spacelike vector is orthogonal to some timelike vector, while true, is not really useful for this case. We already have a better way of picking out which vectors are spacelike--they are the ones tangent to one of the Euclidean 3-spaces, as above--so we don't need orthogonality to pick out spacelike vectors, and since any vector that is not spacelike is timelike, we don't need orthogonality to pick out timelike vectors either.

 

[PeterDonis]

It's the idea that orthogonality is defined by generalizing the idea
that: at a point of circle, the tangent is orthogonal to the radius.

But how do you define a "circle" in Galilean spacetime? There is no metric, so the usual definition of a circle as the set of points that are all at the same distance from the center doesn't work.

Also, if we are talking about boosts, we are talking about vectors, not geometric figures, and the whole idea of a "circle" doesn't really make sense for vectors.

 

[Hornbein]

at least what I see is that the subject of this topic where the - sign comes from seems to be an already assumed to be known starting point in your equation (1.2.1) at the first page. so based on that it cannot help me I assume.

The minus sign is there because our 3+1D universe is hyperbolic. x² - y² = C. It only seems strange because we are used to circles, as in x² + y² = C.

With a circle if |x| grows larger then |y| shrinks. With a hyperbola if |x| grows then so does |y|. We never can complete a rotation in a hyperbolic space. Instead we get closer and closer to an asymptote but never get there. This is more or less why we can't exceed the speed of light.

Traditionally we hear of relativistic length contraction and time dilation, but I find that confusing. With a hyperbola if |x| shrinks then so does |y|. So in my opinion shrinking length goes with shorter time. If the other guy's clock slows down then his time is less, not more. It's exactly the same thing, I just find the traditional label weird. It obscures the hyperbolic nature of the beast. Fortunately there is nothing to prevent me from using the concept of time contraction in the privacy of my brain.

By the way, t² - x² - y² - z² = d² is equally valid and is used by some authors. It doesn't matter where the minus sign is.

 

[robphy]

But how do you define a "circle" in Galilean spacetime? There is no metric, so the usual definition of a circle as the set of points that are all at the same distance from the center doesn't work.

Also, if we are talking about boosts, we are talking about vectors, not geometric figures, and the whole idea of a "circle" doesn't really make sense for vectors.

From my draft at https://www.aapt.org/doorway/Posters/SalgadoPoster/Salgado-TrilogyArticle.pdf
(as part of https://www.aapt.org/doorway/Posters/SalgadoPoster/Salgado-GRposter.pdf
https://www.aapt.org/doorway/Posters/SalgadoPoster/SalgadoPoster.htm
http://www.aapt.org/doorway/TGRU/ )

• as Euclid defines a circle in the plane as
  "the locus of points (say) t=1 km from a point [along radial paths from the point, using odometers measuring t]"
• Minkowski might define the Minkowski-circle (a hyperbola) on a spacetime diagram as
  "the locus of events (say) t=1 s from an event [along inertial worldlines, using wristwatches]".
• in hindsight, let's pretend that Galileo might define the Galilean-circle (a t=1 line) on a position-vs-time diagram as
  "the locus of events (say) t=1 s from an event [along inertial worldlines, using wristwatches]"... just like Minkowski.

as seen in https://www.desmos.com/calculator/kv8szi3ic8

• Vary the $E$-slider: E=-1 for Euclid, E=+1 for Minkowski, E=0 for Galilean.
  (Observe the unit "circles", the radial lines, and the tangent lines.)
• Then, for fixed-E, vary the $v_2$-slider to trace out the "circles".
• Open the "Boost" folder, vary the $v_{LAB}$-slider to see the "circle" get preserved by the boost.
  (That is, the boosted radial unit-vectors remain on the "circle".)

The metric for these affine Cayley-Klein geometries is
$$ds^2=dt^2-Edy^2.$$
(Yes, the Galilean metric is degenerate... one needs another quadratic form for the spacelike vectors,
as is well known. See the papers from Trautman, Ehlers, etc...
See also https://www.physicsforums.com/threads/why-is-minkowski-spacetime-non-euclidean.1016402/post-6647305 and links therein.

While "degeneracy" sounds bad,​

note that from a projective-geometry point of view, Euclidean geometry has degenerate features,​

but there they manage to get along and develop Euclidean geometry.)​

The idea of my approach is to carry out all constructions and calculations keeping track of the $E$,
and to do so in a unified way as much as possible.
A construction which seems difficult to be unified is demoted in favor of one that can be unified.
GOAL: Same storyline, different $E$.

(So, I want to define "timelike", "spacelike", and "null" in a unified way from these structures.)​

The boosts for the above are ( from https://www.physicsforums.com/threads/why-is-minkowski-spacetime-non-euclidean.1016402/post-6648907 )

$$\left( \begin{array}{c} t' \\ y' \end{array} \right)
=
\left(
\begin{array}{cc}
\frac{1}{\sqrt{1-E\beta^2}} & \frac{E\beta}{\sqrt{1-E\beta^2}}
\\
\frac{\beta}{\sqrt{1-E\beta^2}} &
\frac{1}{\sqrt{1-E\beta^2}} & \end{array}
\right)\\
\left( \begin{array}{c} t \\ y \end{array} \right)$$

I learned of these 2-D Cayley-Klein geometries (which include these 9 geometries:
Euclidean, Hyperbolic, Elliptical/Spherical, Minkowski, Galilean, deSitter and anti-deSitter and their Galilean limits) from
"A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity" by I.M. Yaglom
https://www.amazon.com/dp/0387903321/?tag=pfamazon01-20
and, more recently,
Richter-Gebert's "Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry" https://www.amazon.com/dp/3642172857/?tag=pfamazon01-20 .

To see the unified metric for all 9 geometries, look at page 3 of my poster.)​

So, I am carrying out the constructions and calculations,
especially as it applies to physics (PHY101, Special Relativity, and hopefully the deSitter spaces).

 

[PeterDonis]

"the locus of events (say) t=1 s from an event [along inertial worldlines, using wristwatches]"

But this isn't a circle; it's not even a continuous curve. It's two disjoint Euclidean 3-spaces (one 1 second to the future, the other one second to the past), or two disjoint lines if we eliminate 2 spatial dimensions.

 

[PeterDonis]

my approach

This probably needs to be in a separate thread since it seems well off topic for this one.

 

[robphy]

"the locus of events (say) t=1 s from an event [along inertial worldlines, using wristwatches]"

But this isn't a circle; it's not even a continuous curve. It's two disjoint Euclidean 3-spaces (one 1 second to the future, the other one second to the past), or two disjoint lines if we eliminate 2 spatial dimensions.

In the preceding line, I did specify "(a t=1 line)"... not $t^2=1$.
For completeness,
call it the "future Galilean circle" (t=1) as one would get from future-directed inertial observers from an event.
Similarly, for Minkowski spacetime,
we would refer to the "future Minkowski circle" ($\tau=1$) as one would get from future-directed inertial observers from an event.

...and before it comes up... consider only transformations that preserve the direction of time.

 

[PeterDonis]

For completeness,
call it the "future Galilean circle" (t=1) as one would get from future-directed inertial observers from an event.
Similarly, for Minkowski spacetime,
we would refer to the "future Minkowski circle" () as one would get from future-directed inertial observers from an event.

Ah, ok.

 

[malawi_glenn]

ds is no coincidence and should follow in a logical way from the general thoughts behind the whole idea.

Math is logic. It is not coincidence, it follows from the Lorentz-transformations. The postulate was that $(ct)^2-x^2=0$ for light-like separations in all inertial frames. Then if you want to have a transformation that is linear, only depends on $v$ and reduces to Galilean transformation, the result $(ct)^2 +x^2$ is invariant pops out. It is a prediction of the theory. The language of physics is math which is a logical system. What most people mean when they invoke the word "logic" is usually something else than mathematical logic, like "intuition" etc

Remember that I wrote that it is nothing special with t and x,y,z here. As long has you have quantity that transforms as t and quantities that transforms as x,y,z you have the same form of the invariant.
It comes from the geometry which these transformations induce.

Consider the Euclidean space $\mathbb{R}^3$, rotations and translations preserve the Euclidean distance (norm) $x^2 + y^2 + z^2$, thus it is an invariant under those transformations. Would you look for a "logical" reason why $x^2 + y^2 + z^2$ is invariant under those transformations? Well you could argue "its a sphere with fixed radius, of course it will have the same radius if you rotate or translate it

What would the corresponding "sphere" in this non-Euclidean space of $t$ and $x$ be? (we drop the y and z now for simplicity).
What "shape" would be the same in both $S$ and in $\tilde S$? It would not be circles i.e. $(ct)^2 +x^2 = \text{constant}$. It would be hyperbolas $(ct)^2 -x^2 = \text{constant}$
Points lying on the same hyperbola in this space, which we often call Minkowski space, has the same vaule of $(ct)^2 -x^2$ regardless in which reference system it is calculated.

It is something that is true in general and is nice to work out. For a given transformation in a given space. What are the invariants?

Euclidean geometry, but if I understand that well it means not curved

Euclidean geometry you measure distances with Pythagoras theorem. In Newtonian physics, time is not a coordinate, it is a parameter. Therefore the concept of "space-time" is not well defined.