Sparring Sparling II
G. Sparling may like to discuss three time dimensions, but as Louise Riofrio and Alain Connes often like to point out, we should understand how time is emergent, how the arrow of time arises from the expansion of space.
In lessons on special relativity we always use 1-dimensional spaces, because the relative motion of two frames happens along a line, conveniently called an $x$ axis. The Lorentz transformation relates $x$ and $t$ whilst leaving $y$ and $z$ unchanged. The metric considered by Sparling is simply a threefold version of the pair $(x,t)$, namely
$s^2 = x^2 + y^2 + z^2 - t^2 - u^2 - v^2$
where $t$, $u$ and $v$ are all time coordinates. Using imaginary time, which is only proper, the 1-dimensional case looks like the equation for a circle in the complex plane. Similarly, one might interpret the full metric as a 5-sphere in $\mathbb{C}^3$.
But what happened to the speed of light? The metric should really look like
$s^2 = x^2 + y^2 + z^2 - c_{1}^2 t^2 - c_{2}^2 u^2 - c_{3}^2 v^2$
where the $c_i$ form a 3-vector, analogous to the 3-vector for mass in M theory. Deviations from $c_i = 1$ will introduce some ellipticity into the geometry of the hypersurface. Now the homework exercise is to write out $E = m c^2$ three times.
In lessons on special relativity we always use 1-dimensional spaces, because the relative motion of two frames happens along a line, conveniently called an $x$ axis. The Lorentz transformation relates $x$ and $t$ whilst leaving $y$ and $z$ unchanged. The metric considered by Sparling is simply a threefold version of the pair $(x,t)$, namely
$s^2 = x^2 + y^2 + z^2 - t^2 - u^2 - v^2$
where $t$, $u$ and $v$ are all time coordinates. Using imaginary time, which is only proper, the 1-dimensional case looks like the equation for a circle in the complex plane. Similarly, one might interpret the full metric as a 5-sphere in $\mathbb{C}^3$.
But what happened to the speed of light? The metric should really look like
$s^2 = x^2 + y^2 + z^2 - c_{1}^2 t^2 - c_{2}^2 u^2 - c_{3}^2 v^2$
where the $c_i$ form a 3-vector, analogous to the 3-vector for mass in M theory. Deviations from $c_i = 1$ will introduce some ellipticity into the geometry of the hypersurface. Now the homework exercise is to write out $E = m c^2$ three times.
posted by Kea | 11:51 AM
14 Comments:
Well there are some important observations here. The first is that when you split up the speed of light like you did, it makes one consider the possibility that another way of writing this would be to say that there are two speeds of light. The usual, and another speed faster by sqrt(3).
The extra factor of sqrt(3) gives another way of interpreting the violation of the virial theorem seen in astro-ph/0703462.
The virial theorem gives the ratio of potential to kinetic energy. When you modify the speed of light to be sqrt(3)c, you change the natural unit of kinetic energy but not potential energy. Since e = m c^2, you end up with a factor of 3 in the KE, hence the change to the virial theorem.
Also Kea, before I forget, I should mention that it seems that Clifford and I have , but that you might enjoy the New Zealand stunt woman Zoe Bell's character, which is a New Zealand stuntwoman.
Thanks again for a great post. Hee hee, if we used units where c = 1 then Einstein's famous equation would read E = m. We would also have R = t and GM = t. Confusing, but R = t would illustrate how expansion is indistinguishable from the forward flow of time.
Kea, I'm sure you are aware of the bee problem being reported in the news. It seems that I may be involved in it commercially beginning next week, as my buddy is buying some land that has bee hives on it.
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I've read Sparling's paper http://www.arxiv.org/abs/gr-qc/0610068 and it misses the point, it ignores Lunsford's paper and it's stringy in its references (probably the reason why arXiv haven't censored it, unlike Lunsford's). However, it's a step in the right direction that at least some out of the box ideas can get on to arXiv, at least if they show homage to the mainstream stringers.
The funny thing will be that the mainstream will eventually rediscover the facts others have already published, and the mainstream will presumably try to claim that their ideas are new when they hype them up to get money for grants, books etc.
It is sad that arXiv and the orthodoxy in science generally, censors radical new ideas, and delays or prohibits progress, for fear of the mainstream being offended at being not even wrong. At least string theorists are well qualified to get jobs as religious ministers (or perhaps even jobs as propaganda ministers in dictatorial banana republics) once they accept they are failures as physicists because they don't want any progress to new ideas. ;-)
hruumph, your spam filter really SUCKS. I swear this is the last time I type the comment in.
3-D time looks wonderful, can we earmark one direction for free time ?
Anyway, Noether's theorem teaches us there then have to be three separate forms of energy independently conserved. I wonder if the other two can become available when we're over with our reserves of fossil oil...
Cheers,
T.
Hi all. Nigel, your posts are very helpful. Thanks a lot. Yes, Carl, of course that's where this is heading. Moreover, by looking at t as a 3-vector and considering stationary objects in the 'one c' approximation we get the equation R=c(||t||). This is Louise's equation (and Lunsford's), where the usual time parameter is the length of a 3-vector.
Tommaso, I'm so sorry! There's not much I can do about Blogger, and I don't see why you're having problems. I would be very, very sad to see you go (boo hoo).
Indeed it would be a tragedy to lose Tommaso's welcomed comments. One option that comes to mind is to turn off word verification while enabling comment moderation. However, with the apparent popularity of your blog, comment moderation could quickly become a nuisance.
Comment on separate light velocities c1, c2, c3. [I must also say that the strange disappearance of comment just written plus fiddling with these words is irritating!]
In noncompact flat case you can always choose coordinates in such a manner that c1=c2=c3 results (just by scaling). If some time dimensions are compact the situation changes since the length of close timelike geodesic can be taken as the unit of time and defines unique coordinate as angle like variable.
A good argument for additional compact dimensions (in some sense) is the requirement of fundamental unit of length. The size scale of compact "internal" space provides this length purely geometrically.
In string models this kind length unit emerges as dynamical via compactification but Planck length is put in by hand via string tension, which I do not like. All couplings should be dimensionless (and fixed uniquely by some criterion: quantum criticality fixes Kahler couplings as the analog of critical temperature in TGD framework). Doesn't G become dynamical in M-theory?
In TGD CP2 size defines the fundamental unit of length: about 10^4 times Planck length. Imbedding space is given by the God of Number-Theoretically Oriented Mathematicians as the only one allowing Kahler geometry in the world of the classical worlds;-). Physics is unique because of the uniqueness of mathematical existence. Already in the case of loop spaces Kahler geometry is unique but the value of curvature scalar is infinite since infinite-D constant curvature space is in question.
Matti
hey people, don't worry :) I was only talking about the message I was typing in, which was typed two times (as others before), because for some reason the spam filter does not particularly like me, or because I do not correctly decode the letters below the comment box.
Kea is a friend and she deserves my attention since the material she puts on the web is of the highest quality. That goes without saying.
But her spam filter still sucks :)
However, she is not alone on that issue ...
Cheers,
T.
Hmmm. I certainly did not delete that comment, Matti. But comment moderation is too impractical.
Professor George Sparling’s proposal about a six dimensions space-time, dated april 2007, is exactly the same the Italian scientist Enzo Bonacci made during 2006.
Enzo Bonacci’s papers are available on line since January 2006:
http://enzobonacci.supereva.it
Moreover at the end of 2006 he sent his books to all the best universities (Pittsburg’s included) in order to be reviewed.
So there’s no doubt about who’s the father of this new theory…
Luckily Enzo Bonacci is currently being examinated to be awarded in physics for his esadimensional continuum (3 spacelike + 3 timelike dimensions) and further works about matter-antimatter asymmetry, so that many Academicians should have known about his efforts before Sparling’s claiming.
But if the American scientists should manage to steal even this new idea (after many others stolen from Italy, starting by Meucci’s telephone…) I really think that the whole European Scientific Community should complain about such unfair behaviours!
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