Cosmology 101
Louise Riofrio has never given up trying to explain to adults a varying speed of light cosmology that a child could understand. Now, thanks to the theorist Marco Frasca, Carl Brannen has observed that a varying speed of light solution to Einstein's equations results from a five dimensional Kasner metric:
$\textrm{d}s^{2} = -\textrm{d}t^{2} + t(\textrm{d}x_{1}^{2} + \textrm{d}x_{2}^{2} + \textrm{d}x_{3}^{2}) + t^{-1} \textrm{d}x_{4}^{2}$
The mixture of exponents for $t$ arises from the (three dimensionally) isotropic solution conditions
$\frac{1}{2} + \frac{1}{2} + \frac{1}{2} - \frac{1}{2} = 1$
$\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 1$.
These are like magic matrix conditions for a $4 \times 4$ matrix. The fifth dimension, which both Carl and string theorists are fond of, is necessary for these conditions to be satisfied. There is no Dark Force.
Observe that the coordinate speed of light in the three spatial dimensions goes like $c = 1/ \sqrt{t}$. When taken seriously as a solution, Riofrio's equation $R=ct$ then states that universal expansion is characterized by $R = \sqrt{t}$, or $M = R^{2}$. Since this is supposed to approximate a locally emergent cosmology with T duality properties, it indicates that string theorists are wrong to think that mass should correlate with string length, something that every 5 year old quantum mechanic knows.
$\textrm{d}s^{2} = -\textrm{d}t^{2} + t(\textrm{d}x_{1}^{2} + \textrm{d}x_{2}^{2} + \textrm{d}x_{3}^{2}) + t^{-1} \textrm{d}x_{4}^{2}$
The mixture of exponents for $t$ arises from the (three dimensionally) isotropic solution conditions
$\frac{1}{2} + \frac{1}{2} + \frac{1}{2} - \frac{1}{2} = 1$
$\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 1$.
These are like magic matrix conditions for a $4 \times 4$ matrix. The fifth dimension, which both Carl and string theorists are fond of, is necessary for these conditions to be satisfied. There is no Dark Force.
Observe that the coordinate speed of light in the three spatial dimensions goes like $c = 1/ \sqrt{t}$. When taken seriously as a solution, Riofrio's equation $R=ct$ then states that universal expansion is characterized by $R = \sqrt{t}$, or $M = R^{2}$. Since this is supposed to approximate a locally emergent cosmology with T duality properties, it indicates that string theorists are wrong to think that mass should correlate with string length, something that every 5 year old quantum mechanic knows.
posted by Kea | 5:00 AM
7 Comments:
Kea,
Thank you for the link.
Cheers,
Marco
And thank you, Marco, for visiting AF!
Marni,
I guess this new article is very relevant:
http://arxiv.org/abs/0904.1590
Hi Daniel. Gee! So now it's trendy for people to write papers about Riofrio's varying c cosmology?
A critical comment. In general relativistic context one cannot identify c in the proposed manner. The general coordinate invariant statement is that light at the limit of geometric optics moves along light-like geodesics.
For instance, in empty Minkowski space you can introduce standard coordinates
ds^2= c^2dt-dx^2-dy^2-dz^2
and you have standard c from ds^2=0.
You can also go to Robertson-Walker coordinates
a= sqrt(t^1-x^2-y^2-z^2), r= sqrt(x^2+y^2+z^2)/a, theta, phi.
The line element reads in these coordinates as
ds^2= c^2da^2-a^2(dr^2/(1+r^2)-r^2dOmega^2)
The proposed logic would lead to the replacement of with c*a_0/a, a_0 some suitably chosen time value to obtain dimensions correctly, holds true. If you replace a with log(a) you get c again. The correct coordinate invariant definition is in terms of equation ds^2=0 and gives c.
My conviction is that the notion of varying c does not make sense in general relativity. To formulate it one must leave the framework of general relativity and formulate the notion of the measurement of light-velocity as an experimental procedure allowing to compare the values obtained at different points of space-time.
This means basically ability to compare length and time units at different points. Kind of universal length and time units are needed and general relativity does not provide them (TGD does: CP_2 size and p-adic length scale hierarchy and hierarchy of Planck constants).
In TGD universe many-sheeted space-time allows this kind of comparison. Photons can move from A to B along different space-time sheets and you can simply find whether they arrive at B simultaneously or not and what the fractional time lag is (to express it in completely general coordinate invariant manner).
In TGD Universe c is reduced from its standard value because the path along curved space-time sheet is longer than along a geodesic line of M^4xCP_2 with constant CP_2 coordinates. Note that one can also have situations in which photon rotates along geodesic circle of CP_2 so one has light-like geodesic in H but not in M^4.
The light velocity associated with given space-time sheet *increases* in typical standard cosmology which is not surprising since cosmology flattens gradually.
In (possibly local and not so) big crunch the c would *decrease*. This could correspond to a formation of say solar system by condensation of matter to form regions with increasing density of (possibly dark) matter.
Best Regards,
Matti
Hi Matti. We realise that varying c makes no real sense in GR, especially since it entails a variation in hbar.
The problem with letting the speed of light vary is not that it is in violation of experiment; it's that it's in violation of theory. Experiment is always interpreted in the light of existing theory, and since theory holds that the speed of light is constant, that is what experiment "verifies". It's a tautology.
Louise's point is that it's possible to reinterpret the cosmological observations in terms of a varying speed of light. I prefer a version of this where the positions of the galaxies are (more or less) stationary in space. Then the big bang is only an illusion caused by a steady decrease in the speed of light. Things that seem to have been closer together, back towards the beginning of the universe, were actually just about as far apart as they appear to us now; it's just that our measuring stick, light, has changed its speed. (Or equivalently, time is passing much more quickly now so that it appears to us that light has slowed down.)
Making the speed of light variable, and letting distances remain unchanged makes for a much simpler universe than one which was emitted from a singularity. First, the singularity is gone. Second, the bizarre low entropy of the singularity no longer needs explaining. And finally, letting only one thing change, the speed of light or the rate of passage of time, is a far more restrictive (and therefore more predictive) than assuming a singularity.
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