Wednesday, August 30, 2006

Einstein's Field Equation

Just as retrograde motion of the planets shows that Earth is not the centre, the Cosmic Microwave Background is first-hand evidence that c has changed. When we get our heads out of over-analysis and look at it, the average temperature appears the same over large areas. Large regions of Space were able to reach thermal equilibrium. Even at this time of recombination 300,000 years after the Big Bang, the speed of light was much faster.

One book everyone should take to bed (at least until my book is released) is called RELATIVITY: THE SPECIAL AND GENERAL THEORY by someone called Albert Einstein. It lists at only $6.95 US, far less than the $100+ book your professors forced you to buy. No one understood Relativity like Einstein himself. Einstein's mathematical techniques are different and more insightful than the shortcuts used today. The books written by others fail to capture Einstein's understanding of Theory.

In Chapter 31 Einstein attempts to envision the entire Universe. He concludes that the most logical shape is a sphere of 4 dimensions. His theory also predicted that a spherical space would collapse due to gravity. Einstein introduced a "cosmological constant," a repulsive force counteracting gravity. Later he called that his "greatest blunder." We shall not repeat that blunder.

When you have finished that book, move on to THE MEANING OF RELATIVITY for a more detailed mathematical treatment. In the appendix for the second edition, on page 117, he restates the field equation of gravitation, the "Einstein equation."

Here we see that R = ct and GM = tc^3 form a solution to the field equations of gravitation. Any cosmology should solve the Einstein-Friedmann equations. Since this Universe with a changing speed of light also fits the data, there is reason to be optimistic.


Blogger QUASAR9 said...

Louise, some fun tips from
Blue Gal to fellow bloggers

Making blogging fun!
Have a nice day. - Q.

9:13 AM  
Blogger Kea said...

Great recommended reading! I bought an old edition of TMOR once for only $1.

2:06 PM  
Anonymous Anonymous said...

I haven't read Einstein first hand yet; but, thanks to your encouragement, I'll try. My background is electrical engineering and software design (many branes inferior to relativity, I know), but I wonder if Einstein ever considered a non-constant c. And, if not, why not?
Anyway, if there's an "Anything but Dark Matter" club, I'd be tempted to join.


4:16 PM  
Blogger Dimitri Terryn said...


I have a question. Do these types of solutions generalise to more complicated matter distributions? I remember from my cosmology course that the matter in the universe is often modelled as a perfect fluid, does this have the same effect?

I'd check it myself, but I have a elementary particles final tomorrow...

7:02 PM  
Blogger L. Riofrio said...

Hi Kea, everyone! P&M, Einstein DID consider a changing c. The Principle of Relativity page 107: "The principle of the constancy of the velocity of light holds good according to this theory in a different form from that which usually underlies the ordinary theory of relativity."

Dimitri, good luck at the exam. The Friedmann equations are calculated for a spherical mass distribution of average density, within a larger universe that could be flat or curved. Scale factor r is in direct proportion. This mass distribution expands/decelerates at the same rate as the universe.

7:27 PM  
Blogger L. Riofrio said...

P.S. Kea, I've just posted a comment about the speed of light on Not Even Wrong. Let's see if it gets deleted!

7:48 PM  
Anonymous Anonymous said...

Einstein is very readable and clear, which is different from a lot of old-time scientists. The original master is still a master in this century.

8:54 PM  
Blogger Kea said...

Let's see if it gets deleted!

Yup! He deleted Aaron's and my comments, too.

12:28 PM  
Blogger Mahndisa S. Rigmaiden said...

09 02 06

Hello Louise:
since GR isn't my thing, I was wondering if you could explain what each of the variables in your equations mean. I have studied the Einstein equations a little bit, but have found that notation varies greatly from source to source.

Next, your end conclusion is that GM=Tc^3, and that C has varied. Is the M referring to the mass of the universe? And if C changed, does that mean that G has too, according to your theory? What about topological characteristics of your universe? What does your metric look like? You mentioned that Einstein wanted the universe to be spherical, is that what you believe as well?

Are any chaotic solutions to the Einstein equation allowed in your interpretation? Do you allow for roughness at any scale, or is everything isotropic? The WMAP anisotropies mean what within your framework?

Thanks for taking the time to answer these questions!

7:22 PM  
Blogger Mahndisa S. Rigmaiden said...

09 02 06

I should have asked if everything is smooth in your interpretation, not isotropic;)

11:06 PM  
Blogger L. Riofrio said...

HI Mahndisa, always great hearing from you. The many terms are indeed confusing, especially since Einstein used German notation. Some terms represent 4 x 4 matrices: R is the Ricci tensor, g is the spacetime metric, T is the stress-energy tensor. Rho is density and z is an integration constant. Kappa is another constant that normalises to 8\pi G in the Friedmann equation. Here r is the scale factor. That is probably not enough explanation, so I encourage everyone to get those books.

M is mass of the entire Universe, which is nearly constant at this late age. Dirac and Eddington proposed varying-G cosmologies, but those have been ruled out by multiple experiments. I agree with Einstein that it should be spherical, as illustrated in the August 8 blog entry. The metric of Special Relativity is the equation for a sphere.

Initially the Universe may have been nearly isotropic. Today's Universe is smoother than a billiard ball, but on our scale there are many anisotropies shown by WMAP. These are born from quantum fluctuations grown large by expansion of Space/Time. Many of these fluctuations collapsed into singularities, seeding formation of clusters, galaxies and even smaller structures.

Your comments are full of ideas for things that should be expanded upon in the future.

9:07 PM  
Blogger Mahndisa S. Rigmaiden said...

09 04 06

Thanks Louise:
What about chaotic solutions or roughness of the universe via LQG or fractal discretizations of the spacetime? You don't buy those ideas? Thanks for the time you took to answer my queries. BTW I cannot believe Steve Irwin is dead. Damn!

5:23 PM  
Blogger L. Riofrio said...

Hi Mahndisa: The work done by Lee and others on LQG is fascinating, and may provide an additional way to link GR and QM. It is possible that someday we will find that Space/Time is discrete and not continuous. I don't consider LQG to be a "rival" idea, but something to be encouraged.

8:26 PM  
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