Minkowski Space Time Briefly Revisited

Alternative Theories?

 

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Unless, rpenner jumps in to make some sense of it, it does apear so!

 

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Of course they are "real" in that the changes of the physical description due to an imaginary choice of a different rest frame are exactly the changes of physical description due to a real choice of change of state of motion of the real object being considered. If things can have their state of motion changed, then understanding the Lorentz transform is critical to doing physics.

You have confused in the below description the Lorentz transform, sometimes called a hyperbolic rotation, with a Terrell rotation. A Lorentz transform changes the physical description of the dimensions of physical objects; A Terrell rotation changes the visual depiction of moving objects due to a combination of finite propagation rate of optical signals and the Lorentz transform.

See: http://faraday.physics.utoronto.ca/PVB/Harrison/SpecRel/Flash/ContractInvisible.html

You have a chip on your shoulder, but that doesn't make you gifted at physics.

Your story lacks corroborating details, in that you fail to describe how anyone passed the course if the instruction was defective.

 

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Consider 2 bodies in relative uniform motion with velocity \(v\) along their common \(x_3,\,x'_3\) axis.

We know that then
\(x'_1=x_1,\,\,\,x'_2=x_2,\,\,\,x'_3= \gamma (x_3-vt),\,\,\,t'=(\gamma (t-\frac{vx_3}{c^2})\)

where \(\gamma = (1-\frac{v^2}{c^2})^{-\frac{1}{2}}\)


Write \(x_4 \equiv ict\),so for the Lorentz transformation \(L\) and the 4-vector v one has that \(v'=Lv\),

Let \(L=\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&\gamma & i\frac{v}{c}\gamma\\ 0&0&-i\frac{v}{c} \gamma& \gamma \end{pmatrix}\) which yields the transformation above

Now set \(\gamma \equiv \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}= \cosh \alpha\) so that \(\cosh \alpha\) ranges from \(1,\,\,(\frac{v}{c}=0)\) to \(\infty,\,\,(v=c)\)

So that \(\sinh \alpha = (\cosh^2 \alpha-1)^{\frac{1}{2}}=\frac{v}{c}\frac{1}{\sqrt{1-\frac{v}{c}}}\). So the matrix \(L\) becomes

\(L \alpha= \begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0& \cosh \alpha &i \sinh \alpha\\ 0&0&-\sinh \alpha& \cosh \alpha \end{pmatrix}\)

If that's not a hyperbolic rotation, I wold like to know what is

 

There are many ways to write a rotation matrix:

\(\begin{pmatrix} 1 & -a \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ \frac{2a}{1 + a^2} & 1 \end{pmatrix} \begin{pmatrix} 1 & -a \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} \frac{i}{\sqrt{2}} & -\frac{i}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix} \begin{pmatrix} e^{-ib} & 0 \\ 0 & e^{ib} \end{pmatrix} \begin{pmatrix} - \frac{i}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{i}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix} = \exp \begin{pmatrix} 0 & c \\ -c & 0 \end{pmatrix} = \begin{pmatrix} \cos d & \sin d \\ -\sin d & \cos d \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{1 + f^2}} & \frac{f}{\sqrt{1 + f^2}} \\ \frac{-f}{\sqrt{1 + f^2}} & \frac{1}{\sqrt{1 + f^2}} \end{pmatrix}\)


There are strictly analogous ways to write a Lorentz matrix.

\(\begin{pmatrix} 1 & -g \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ \frac{-2g}{1 - g^2} & 1 \end{pmatrix} \begin{pmatrix} 1 & -g \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix} \begin{pmatrix} e^{-h} & 0 \\ 0 & e^{h} \end{pmatrix} \begin{pmatrix} - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix} = \exp \begin{pmatrix} 0 & j \\ j & 0 \end{pmatrix} = \begin{pmatrix} \cosh k & \sinh k \\ \sinh k & \cosh k \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{1 - \ell^2}} & \frac{\ell}{\sqrt{1 - \ell^2}} \\ \frac{\ell}{\sqrt{1 - \ell^2}} & \frac{1}{\sqrt{1 - \ell^2}} \end{pmatrix}\)

But I don't think that is danshawen's stumbling block.

 

rpenner said:

"A Terrell rotation changes the visual depiction of moving objects due to a combination of finite propagation rate of optical signals and the Lorentz transform."

Indeed, and some most impressive mathematical representations for it, too. Good choice. I'm a big fan of Roger Penrose as well.

Farsight gave us a nice set of graphic depictions of what such rotation / distortion does to a sphere, a few posts back.

But I changed those "tail lights" into pulses for a reason. They are no longer "objects", do not have curved surfaces, and as a matter of fact, they no longer "rotate" in any mathematical or physical sense. They appear only to shift with respect to each other, based on differences in simultaneity, which is a much simplified Lorentz problem not even requiring boost matrices.

And why should we even care about the propagation delay of light in such depictions? The question I was most interested in here was, does the object (or the pulses), in any real physical sense, undergo some sort of physical rotation? It's kind of important to know this, particularly if your interest is in whether or not rotation is a possible mechanism for energy to become bound. I think that rotation is a candidate for this function, but you'll never get there with Minkowski's very flawed math. I really don't care how the thing might "appear".

You've already demonstrated that in the case of pulses, time dilation shows up only as Doppler shifts, and in one case in which it does for matter, length contraction doesn't evidence for energy pulses at all. Don't backpedal now and defend Minkowski rotation on grounds that actually have nothing to do with relativity or physical reality. It deserves no such defense.

 

Not this time. I'm just taking an existing theory (Special Relativity sans Minkowski) to its logical conclusion.

Minkowsky may have pulled higher dimensions out of his (or Euclid's) mathematical hat, but I've demonstrated how to synthesize them from a deeper understanding of dimensions that can actually be observed. Show me that space = c * time is not a basic tenet of relativity, and you can dump it into alternate theories.

 

I really must have missed how you got to the above conclusion.

 

Dan, speed = distance divided by time, which obviously implies distance = speed times time is a tenet of all physics, from the Greeks onwards, not just relativity. Hell, I learned that aged 8

Of course, if you want to apply this to the spacetime 4-manifold, you had better know what you mean by "distance". For that you need a metric, whose existence you may not assume. As it happens, in GR, the metric is key, because from it we can extract the connection from which in turn we can extract the curvature.

Of course, this is just "mathematical nonsense" according to you........

 

Because space is three-dimensional and time is one-dimensional, the notation "space = c × time" is nonsensical.

Because distance = speed × time is part of the description of phenomena that predates Newton, it is not special to relativity.
But because not all phenomena propagate a the speed of light, "distance = c × time" has no universal applicability.

What is innovative and universal in special relativity is that \((c \Delta t)^2 - (\Delta \vec{x})^2 = (c \Delta t')^2 - (\Delta \vec{x}')^2 \) for two descriptions of the same events with respect to a different inertial standard of rest. This implies a certain relationship between coordinate time and physical elapsed time, meaning time was not absolute. This also implies \((c \, \Delta t + \Delta x_{\parallel})(c \, \Delta t - \Delta x_{\parallel}) = (c \, \Delta t' + \Delta x'_{\parallel})(c \, \Delta t' - \Delta x'_{\parallel})\) for the component of \(\Delta \vec{x}\) which is parallel to the difference of rest motions of the two standards of rest. So light-like motion has a particular physical significance in special relativity.

Also, when written in physical units where c is unitless and of magnitude 1, the Lorentz transform is always a symmetric matrix of determinant 1:

\( \begin{pmatrix} \frac{1 + g^2}{1 - g^2} & - \frac{2g}{1 - g^2} \\ - \frac{2g}{1 - g^2} & \frac{1 + g^2}{1 - g^2} \end{pmatrix} = \begin{pmatrix} \frac{e^{-h} + e^h}{2} & \frac{-e^{-h} + e^h}{2} \\ \frac{-e^{-h} + e^h}{2} & \frac{e^{-h} + e^h}{2} \end{pmatrix} = \begin{pmatrix} \cosh j & \sinh j \\ \sinh j & \cosh j \end{pmatrix} = \begin{pmatrix} \cosh k & \sinh k \\ \sinh k & \cosh k \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{1 - \ell^2}} & \frac{\ell}{\sqrt{1 - \ell^2}} \\ \frac{\ell}{\sqrt{1 - \ell^2}} & \frac{1}{\sqrt{1 - \ell^2}} \end{pmatrix}\)

where \(\ell = \tanh j = \tanh k = \frac{-e^{-h} + e^h}{e^{-h} + e^h} = - \frac{2g}{1 + g^2}\) is called the velocity parameter, \(h = j = k = \ln \sqrt{ \frac{1 + \ell}{1 - \ell} } = \tanh^{-1} \ell = - 2 \tanh^{-1} g\) is called the rapidity parameter, and \(g = - \frac{1 - \sqrt{1 - \ell^2}}{\ell} = -\tanh \frac{j}{2} = -\tanh \frac{k}{2} = \frac{1 - e^h}{1 + e^h}\) is a parameter which corresponds to -1/2 of the rapidity.

 

Is this about length contraction and time dilation?

 

Very much so, and also about energy, momentum, inertia. Not at all about shape (or topology), and that's the point.

I really, really wanted to get that point across on the last round of Minkowski, but somehow it never seemed to fit.

Thanks for the assist, BWS.

 

One dimension that is time makes perfect sense for energy. Stipulate that it can propagate in any direction and you have 4D space-time, given that c is a dimensionless constant relating space to time, the way Einstein always intended; not a corruption of physical reality to conform to Euclidean space with time thrown in as an afterthought, the way Minkowski intended.

Are you saying that any length ('space') cannot be expressed as light travel TIME? This is relativity 101.

You are helping this idea evolve. When we started out, I was clueless about whether space, time were "granular". Now we actually know that if one is, the other must be. This relationship is why.

 

I don't believe much if any of what you just stated has been demonstrated acurate, let alone consistent with reality.

 

So, what's the alternative? Space that is just like Euclidean space, with an origin, uniform rate clocks, three mutually orthogonal axis marked in units from minus infinity to plus infinity?

I'm certain I don't need to remind anyone on this thread that Euclidean space is not somewhere that relativity (or reality, or even your GPS) actually works. The manner in which time is introduced to the model makes a big difference.

A light year is an FoR invariant way to describe the distance light travels in a year. This came from relativity 101. A light inch or centimeter is the same idea. NIST bases its metrology science on the number of wavelengths of an optically pumped rare Earth to define the length of a meter, and this is why. If you really need references, I can provide them. I managed an electronic calibration laboratory for part of my career.

 

That's all garbage and misdirection.

Minkowski spacetime was developed to better describe special relativity, which is used everyday in labs like the LHC. It is locally consistent with General relativity. And it is only within the larger context of general relativity that the GPS system issues become an issue at all. By that time you are no longer constrained by the earlier Minkowski spacetime, again intended to better describe/explain the flat Spacetime of SR.

The whole meter and time based on the speed or frequencies of light Farsight crap is a dead horse. Both the second and the meter had been established long before the existence of atomic clocks and accurate measurements of the speed of light. When our equipement and devices become accurate enough to measure hundredths or thousandths of an inch or even centimeter, we did not suddenly claim that the inch/centimeter, was based on these new mechanisms of measurement. You are playing a shell game, of words and mixed-matched concepts.

IOWs you're crackers. Both special and general relativity, as well as Minkowski spacetime are safe and accurate within the scope of the applicability of each. And no, no flat Spacetime coordinate system accurately describes gravity as described within the context of GR. That does. Ot make it an invalided or even outdated system.

 

Before relativity, there was nothing routinely referred to as light travel time in physics. As far as any classical physicist was concerned, a solid gold meter stick kept in a bell jar at standard temperature and pressure was as good a standard of length as any other. After relativity, it was realized that small variations not having anything to do with temperature, pressure, or anything with which classical physics was familiar with could surreptitiously tamper with their vaunted standard, and so standards that were based on the frequencies light emitted from the incandescence of stable isotopes of precious metals became the norm. After lasers were developed, the standard of length changed again. I'm not saying that the history of standard weights and measures always reflected the best available technology, or that other sources of error did not creep into the discipline from unanticipated directions. But these guys, particularly today, are very keen to keep measurement in step with the latest scientific knowledge. We depend on these instruments in order to advance physics, among other things.

Classical physics has a robust set of laws of rotational dynamics analogous to the ones Newton devised for position, velocity, acceleration, which work wonderfully well whenever and wherever rotation is found. I'm saying that Minkowski space-time rotation is in no way part of that. The structure of it doesn't even hint that it might be a real rotation; only that it is a rotation requiring higher dimensions to understand. This sort of relativistic rotation is neither important nor essential to other relativistic calculations, even in small measure.

So the question I had was, is this really like a rotation? How much rotation is even possible? Could the automobile headlights rotate as far as, say, 90 degrees at some increased relativistic speed? And then the point of the two pulses of light separated by the distance of the tail lights is clear. If anything is going to actually rotate with this model, something traveling at c (FOR REAL) would be maximal. This is an actual original thought experiment. I'm not playing Farsight's whacko advocate, OnlyMe, but I do understand your concern.

Understanding that Minkowski rotation was of no use to understand the relativistic or other dynamics of bound energy that I am working with enabled me to take a fresh look at the way it was put together. I think that what I came up with is an improvement in many respects, but if you don't, that's OK too. I'm not crackers enough to get upset about it.

 

OK, you will disagree with the following (even though you won't understand it).

As I showed, as rpenner showed, the issue is not a rotation of a "real physical object" rather it describes rotational symmetry of a rather special kind. It is called \(SO(1,3)\) or rather, since this is a set (in fact a group) of 4 rotational symmetries, each rotational symmetry is a group element

 

Just for fun,
\(\begin{pmatrix} \textrm{sech} m & \tanh m \\ - \tanh m & \textrm{sech} m \end{pmatrix} \begin{pmatrix} A \\ B \end{pmatrix} = \begin{pmatrix} A \, \textrm{sech} m \; + \; B \, \tanh m \\ B \, \textrm{sech} m \; - \; A \, \tanh m \end{pmatrix} , \quad \left( A \, \textrm{sech} m \; + \; B \, \tanh m \right)^2 + \left( B\, \textrm{sech} m \; - \; A \, \tanh m \right)^2 = (A^2 + B^2)(\textrm{sech}^2 m + \tanh^2 m) = A^2 + B^2\).
So this is a rotation matrix in disguise. Does it follow that \(\begin{pmatrix} \sec n & \tan n \\ \tan n & \sec n \end{pmatrix} \) is a hyperbolic rotation?

In post #103 of this thread, I demonstrated that the same Lorentz transform which relates <L, v> to <L', v'> (in post 95) also must relate <E, p, v> to <E', p', v'> so it seems Lorentz transforms (hyperbolic rotations) have fundamental physical utility.

\(v' = \frac{v - u}{1 - \frac{u \, v}{c^2}}\)
\(L' = \frac{\sqrt{1 - \frac{u^2}{c^2}}}{1 - \frac{u \, v}{c^2}} L \)
\(p' = \frac{p - \frac{u}{c} \sqrt{p^2 + m^2 c^2}}{\sqrt{1 - \frac{u^2}{c^2}}}\)
\(E' = \sqrt{ c^4 m^2 + \frac{(p c^2 - u E)^2}{c^2 - u^2}}\)

Because motion is relative, describing one inertial object in many frames of coordinates is the same as describing identical objects in many states of inertial motion in the same frame of coordinates. Thus study of the relation between frames of coordinates is essential. Thus study of the Lorentz transform is an appropriate topic in physics.

 

Thanks for updating the formulas, rpenner. I'll research. If you can think of anything else salient to the topic of the thread, please indulge. I think this more than answers my questions.