Discussion between BuddyJohn and sweetser - 2017-03-12

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1:16 AM

The chrome extension worked perfectly, thanks.

My goal is to treat events in space-time as numbers, rank zero tensors. Real and complex numbers are too small, but quaternions are just the right size. The difference between two events squared is: $(Dt, Dx, Dy, Dz)^2 = (Dt^2 - (Dx^2 + Dy^2 + Dz^2), 2 Dt Dz, 2Dt Dy, 2 Dt Dz)$. Note that I am using a capital D to indicate this is a discrete difference, not a differential.

The first term has the same form as the Lorentz invariant interval of special relativity. If two observers are looking at two events and are traveling at a constant velocity relative to each other, they will agree to the first term of the quaternion difference squared. This quantity will not change if the observer is translated somewhere else in space, nor if they are rotated.

The other three terms I don't think even have a name in physics. Do you know of a label for $tx, ty, tz$? I now call them space-times-time. These will be different for these two observers.

What I have been studying recently is what physics results when the observers disagree about the discrete interval but agree on the space-times-time? I believe this symmetry could be all that gravity is about. The math I am trying to live without is that which uses metric tensors. In its place is number theory. The justification is a new approach to gravity.

If true, then like special relativity, there would be a symmetric and nothing else, no field theory per se so no particle, no graviton. I know that will make me an unpopular kid.

Note: I should have written for the space-times-time terms $Dt Dx, Dt Dy, Dt Dz$. It is vital to use discrete differences. I did not see a way to edit an earlier comment.

1:40 AM

I don't think you've really internalized the lessons from above.

For instance: "The first term has the same form as the Lorentz invariant interval of special relativity."

That is only true for inertial frames with Cartesian coordinates.

It wouldn't even work for other coordinate systems.

It is not hard to make it work for other coordinate systems so long as one always uses the Minkowski metric in Cartesian coordinates. Make all your measurements in terms of $R, \theta, \phi$ and plug into the first position $R \cos \theta \sin \phi$, $R \sin \theta \sin \phi$ in the second slot, and $R \cos \phi$.

People like using metrics that change in their location, but that is a technical choice people are making, one with a long tradition as you note.

One could claim I was using one metric choice, except I am not using a metric tensor, like ever.

So this is like an Aether theory. You can use other coordinates, but you need to convert to the "real" coordinate system before doing any calculations.

You don't even have rotational symmetry.

Of course there is rotational symmetry.

There are no linear transformations that preserve $DtDz,DtDy,DtDz$ except for translations, and that is already a symmetry of special relativity.

I have no idea why you bring up Aether theory. It is number theory I am trying to use. There are many ways to represent numbers, but only one rule for adding and multiplying them.

Think of complex numbers. There is the Euclidean form and the polar form. The numbers might be different, but one always uses the same rules to multiple the real and complex parts.

1:56 AM

Do you know what Lorentz's aether theory was, and how it related to special relativity?

It is admittedly an aside, but the historical lesson may be useful here.

I don't want to deal with that historical lesson, sorry.

ok

I do want to deal with the case of complex numbers.

Okay. Let's say we live in a two dimensional space-time

Make measurements in the complex plane with x and y. Repeat with $R$ and $\theta$. Take the square.

$z^2 = (x, y)^2 = x^2 + y^2$

2:02 AM

You wanted to stick to affine spaces, remember.

You cannot use polar coordinates.

Arg, messed it up.

doesn't matter, I know what the square looks like

$z^2 = (x, y)^2 = (x^2 - y^2, 2 x y)$

Actually, just to be clear, you are not talking about multiplying events in space-time. You are still talking about discrete differences, right?

yes

The way I will deal with the affine issues is to measure $R$ and $\theta$ and always do calculations with $R \cos \theta$ for the real part and $R \sin \theta$.

I realized that looks like cheating, not using an interesting metric in a way.

But I think it is a choice I can make.

2:08 AM

Exactly, you are favoring one representation over another. As you must, because you are trying to work with an affine space.

I am fine with that. I am really making the measurements with $R$ and $\theta$.

Even more, you (I think?) want to interpret $Re(Dz)$ as a distance measurement, and $Im(Dz)$ as a time measurement. So you already have a a portion of a metric implied here.

Yes and no. Yes to $Re(Dz)$ as a distance measurement. The $Im(Dz)$ is space-times-time, so a mixture of the two that few - even myself - have used.

No. Or maybe you had a typo.

Oh, I see, you meant $Im(Dz^2)$.

I haven't gotten to that yet.

rught, sorry

So $Re(z)$ is time, $Im(z)$ is space, $Re(z^2)$ is a distance, $Im(z^2)$ is a space-times-time.

2:18 AM

oh okay, I put it backwards. I guess the sign is kind of arbitrary in 2D, but to allow your expansion to quaternions later, that makes sense.

Please insert all the D's for discrete differences.

Actually, do you mind if I pause to try to summarize up to this point to make sure we are on the same path?

k

The events in this two dimensional space-time are represented by the complex numbers.

The "correct" representation is a Cartesian one $z = t + i x = (t,x)$ where $t$ and $x$ are real numbers.

If we label events with other two tuples of real numbers, we first convert to the correct representation (or if we know the direct mapping to a complex number, use that), so that we can then calculate with the complex numbers.

Why I was calling it an Aether theory, is that another observer will somehow need to know his velocity relative to the "background" or "correct" representation in order to convert his tuples to the actual complex number.

Are the first three sentences okay? And I assume you feel I missed something going to the fourth.

A minor tweak on the first 3: better to use $z = t + i R$ so it is a smaller notational jump to quaternions. All that happens is $i$ goes to $i, j, k$.

And I do want to get the delta in soon so an origin does not have to be agreed upon by all observers which we know would be a crazy bad nightmare.

One other idea I think of early on is that different observers usually have different $Dz$'s. Things only start to get interesting with the square.

2:33 AM

Wait, what? How can different observers have different Dz's?

They are in relative motion to each other.

Doesn't Dz mean the discrete difference of the complex numbers for two events?

So we already converted to the complex numbers. There should be no disagreement.

I thought this was basic special relativity. The values for $Dt$ and $DR$ will both be either larger or smaller in such a way that $Re(Dz^2)$ remains the same.

Let's look at one event, and call it A. I thought your idea was that different observers might have a different tuple representation for that complex number, but it would be the same complex number.

You said: "It is number theory I am trying to use. There are many ways to represent numbers, but only one rule for adding and multiplying them."

Indeed, but this is mathematical physics. There is one rule for multiplication. But the two observers are in motion relative to each other.

2:40 AM

You are skipping over something important here, because you are not making sense.

If two observers are in motion relative to each other, wouldn't that just mean they might choose a different tuple representation? But ultimately, event A is still the same complex number.

...

Are you saying the different observers will not even agree on what complex number event A is?

First, I need two events, A and B and their difference, AB. Observer M is not moving towards either. Observer N is moving towards AB. N will say the time between AB is less, and the distance is shorter.

Are you saying the different observers will not even agree on what complex number event A is?

The only thing they agree on is $Re(Dz^2)$.

$t^2 - R^2$

$Dt^2 - DR^2$

So the points in spacetime are not associated with a complex number in your idea?
Instead *each* observer independently chooses how to label events with complex numbers? That is HUGE difference.

what happenned to it just being different representations of the same complex numbers.

We talked about the Euclidean versus Polar representations. There one always write the polar variables in a Euclidean form.

There certainly is no background complex plane that everyone agrees to use. There is no agreement on where the origin is for example. The origin makes a difference for describing $A$ but not $D(AB)$.

2:58 AM

"There certainly is no background complex plane that everyone agrees to use."

I thought you said the opposite earlier.

Well, you had read Aether theory into my words earlier, but that was never my intent.

The rules for addition and multiplication are fixed, not what direction R is or the location of the origin.

If each observer is independently allowed to decide what complex numbers are associated with events, then why can't an observer take any two-tuple coordinate system (u,v) and choose z = u + i v

If anyone wishes to compare their results to (u, v), they must use their system so an apple can be compared to an apple.

So observers can't independently choose what complex numbers are associated with events? Only one gets to choose?

You seem to be trying to have it both ways.

Yes. One because the reference number system. It could be Euclidean, it could be polar.

One becomes...

This is how I visualize it. Make 2 does on a clear piece of acetate. Now put underneath a square graph paper. Good old Euclidean coordinates. Now swap that with polar graph paper. The tuples all change, but the distance between the two should remain the same.

3:13 AM

See, now you are going back to arguing it is different representations but same complex numbers.

So why isn't another observer's choice of tuples just another representation of the same complex numbers?

Okay, I skimmed this whole discussion again.

I am sorry, your question is confusing to me. I sometimes think I should just say f--- it, impose Euclidean law on measurements.

It seems like the cartesian representation is important to your idea.

yup

Okay, let's just start there.

It seems like the jump to complex numbers is actually secondary then.

We have this 2D linear space, and we are considering the different Cartesian coordinate systems we can describe this with, and then associating each of those with a map of events -> complex numbers using z = t + ix.

I need the multiplication rule of complex numbers.

k

3:26 AM

And then of these maps, some subset will agree on Re(Dz^2) for any two events.

yup. And there is a little physics to that subset

And so I'll just guess you want to postulate that physics described using any of those coordinate systems will appear the same.

Is that the point?

It is the physics of special relativity. So the folks in the subset agree on $Re(Dz^2)$. If they also agree on $Im(Dz^2)$, then there is no relative motion between the two. if the disagree, there is motion and the Lorentz transformation can be sued to convert between the two observers under study.

Not quite.

You don't have enough to get SR.

For instance Galilean relativity still fits with this.

I did not think a Galilean transformation would leave $t^2 - R^2$ invariant.

3:34 AM

This is something easy to test. Try it out.

$t' = t$, $x' = x - vt$. $t^2 - (x -vt)^2 = t^2 -x^2 - 2xvt - v^2 t^2$

Did I make a math error?

At the very least, that should be a +2xvt

A delta x would be the same: $x'_1 = x_1 - v t$, $x'_2 = x_2 - v t$, $dx = x_2 - x_1$

3:54 AM

Sorry for the slow reply. You are correct, and it seems obvious now. I am currently trying to figure out my error of intuition.

It is the speed of light being the same for all inertial observers that break the Galilean ceiling.

The Galilean transformation leaves $Dt^2 - DR^2$ invariant, not $t^2 - R^2$, but nobody cares about $t^2 - R^2$.

Ah, okay, I see my brain fart now. In Galilean relativity, there is an absolute time, and people will not disagree on the lengths of objects.
I mistakenly jumped to that meaning Dt^2 and Dx^2 will individually be agreed on.
And while that works for Dt^2, agreeing on the lengths of objects is not general enough to say Dx^2 is agreed on.

So do you know think agreeing on $Dt^2 - DR^2$ is enough to get SR? That was how I was taught special relativity by Edwin Taylor, author of "Space-time Physics".

Those are the functions one uses to create the light cone and all its hyperbolas.

Well, for SR we also need to associate these with inertial motion. But sure, sounds good.

k. Need to go to sleep. Thanks for the chat. And if you get a chance, in your own terms, go ahead and play with $Im(Dz^2)$. It is an unusual cat I believe.

4:06 AM

k

My email address is widely available. Spam filters work well enough for me.

But I will follow your advice which has been good.

To be honest, I didn't think the chat would last this long. I have no idea how you think this will lead to gravity without any fields.
You have got me curious now, and I am willing to chat a bit more later. Maybe I'll try to extrapolate what we did here to Im(Dz^2), and I'll likely make different assumptions and get it wrong, but you can correct me when you get the chance.

1 hour later…

5:33 AM

Alright, I read through the discussion again. (Okay, I admit I mostly skimmed.)

Some issues:
My comment here is wrong. http://chat.stackexchange.com/transcript/message/35980221#35980221
"There are no linear transformations that preserve $DtDz,DtDy,DtDz$ except for translations, and that is already a symmetry of special relativity."

As there are some scaling type transformations that also work.

And while my comment here sounds okay: http://chat.stackexchange.com/transcript/message/35981721#35981721
"And then of these maps, some subset will agree on $Re(Dz^2)$ for any two events."
We should not forget that there is more than one such subset. These different subsets are basically related to each other by scaling of what is a unit length or time.

One more note, the transformations that leave $Dt^2 - Dx^2$ invariant is actually a slightly larger symmetry than SR. Both time reversal and parity leaves that invariant. So those need to be ignored.

6:32 AM

Okay, let me start by trying to rephrase what you introduced last time. Separating what are just basically definitions, from what you are actually postulating.

..

Consider a two dimensional space-time.

Assume observers have access to standard rulers and time pieces.

For a coordinate system $x^\mu = (t,x)$, define $Dx^\mu(A,B) := (t_B - t_A, x_B - x_A)$ or if clear from context, just $Dx^\mu$. If a coordinate system meets some requirements (stated in a bit), the observer can associate a complex number with each space-time point, $z = t + i x$, and similarly for $Dz$ defined from $Dx^\mu$, the displacement between two events.

..

The requirements for a coordinate system to have a complex number representation in this idea are as follows:

* for a clock sitting at the same spatial coordinates, $Dz$ between any events on the clock is the time measured by the clock

* for a ruler at rest, $Im(Dz)$ for any events on opposite ends of the ruler, is the distance measured by the ruler

* remote time measurements would agree with Einstein clock synchronization (or if you don't want to refer to light yet, you can use some equivalent like the limit of slow clock transport en.wikipedia.org/wiki/Einstein_synchronisation )

(Note: I think this fixes the scaling issue, excludes non-Cartesian coordinates, and garauntees some assumptions made about the components.)

..

With that background explained, my understanding is that your idea is:

Consider the subset of complex coordinate systems which agree on $Re(Dz^2)$ for all pairs of events.

These are postulated to be inertial coordinate systems and physics described using any of those coordinate systems will appear the same if they agree on time and spatial orientation (to prevent parity issues).

..

Hopefully I didn't butcher it too much.

The statement of your idea is simple, once the background stuff is all defined.

2 hours later…

8:12 AM

On second thought, it may be that with the above background explained, you may not have to assume these are inertial coordinate systems.

Its possible the restrictions make it so the coordinate systems which meet the requirements to have a complex number representations are inertial coordinate systems.

I've wondered about this before, how restrictive common assumptions about coordinate systems are. I've put it as a question now on the site:
http://physics.stackexchange.com/questions/318193/can-only-inertial-coordinate-systems-meet-these-requirements

I've tried to make it clear enough to not get down-voted. We'll see.

10 hours later…

6:20 PM

I am striving to the to the "On second thought,..." idea. In GR, there are two different ways to get to a "flat" metric. One has the ratio of M/R go to zero. The second is to just be very local. I don't know how to relate that to my own effort, but may be worth thinking about.

Nearly all experimental tests of special relativity happen here on Earth which has both a weak gravity field and involves measurements that are "local".

I up-voted the question.

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Mar '1712

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