LaTeX at last ! (Illustrated with a GR calculation) Sunday, Feb 18 2007

After waiting for long, Blogphysica gets $\LaTeX$ . (See the announcement at the wordpress.com Blog )

The Basic syntax is $latex <LaTeX Equation>$. You can find a list of $\LaTeX$ symbols here(pdf) . In particular, if you find the formulae are too small try $latex {\displaystyle <LaTeX Equation>}$ . It’s great !

Update(18/02/07) : See this FAQ for some more options.

Now, to illustrate LaTeX, I’ll take up a particular problem. Consider two equal masses falling towards each other, as shown below, starting from rest.

m—>—o—<—m

The question is this – How much power does this system lose as the two masses fall towards each other ?

Let $r$ be the distance from the centre of mass (which I’ve denoted by an ‘o’ above and I choose it to be the origin). Take the common axis to be z-axis.

Hence the position of two masses are respectively $(x,y,z)=(0,0,r)$ and $(0,0,-r)$

To the zeroth approximation, Newtonian mechanics tells you that

${\displaystyle \frac{d^2 r}{dt^2} = - \frac{G_Nm}{(2r)^2}}$

(We neglect the effect of gravitational wave on the masses)

The quadrupole moment $Q_{ij}$ of a mass distribution is defined by

${\displaystyle Q_{ij}(t) = \iiint d\forall\ \ \ \rho(x,t) \left[ x_i x_j - \frac{1}{3} \delta_{ij} r^2 \right]}$

Where the integral is done over the whole source($\rho$ is the mass density of the source). It is basically negative of the traceless part of the moment of inertia.

The Einstein formula for the power emitted by the source (in the form of Gravitational waves) is

${\displaystyle P = \frac{G_N}{5c^5} \sum\ \frac{d^3 Q_{ij}}{dt^3}\ \frac{d^3 Q_{ij}}{dt^3} }$

where the symbol $\sum$ denotes a sum over $i,j=1,2,3$.

Assuming that the masses are small in size, the components of quadrupole moment in this case are

${\displaystyle Q_{zz}= m\left(r^2 - \frac{1}{3}r^2\right)+ m\left((-r)^2 - \frac{1}{3}r^2\right)= \frac{4}{3}mr^2}$
${\displaystyle Q_{xx}=Q_{yy} = m(0^2 -\frac{1}{3}r^2)+ m(0^2 -\frac{1}{3}r^2)= -\frac{2}{3}mr^2 }$
${\displaystyle Q_{xy}=Q_{yx}=Q_{zx} =\ldots = 0}$

Now to calculate the third time derivative, we first use chain rule to get

${\displaystyle \frac{d^3}{dt^3} r^2 = 6 \frac{dr}{dt}\ \frac{d^2 r}{dt^2} + 2r\ \frac{d^3 r}{dt^3} }$

We can now employ Newton’s Law to get

${\displaystyle \frac{d^3}{dt^3} r^2 = - \frac{G_Nm}{2r^2}\ \frac{dr}{dt}}$

${\displaystyle \frac{d^3 Q_{zz}}{dt^3} = - \frac{2G_Nm^2}{3r^2}\ \frac{dr}{dt}}$
${\displaystyle \frac{d^3 Q_{xx}}{dt^3}=\frac{d^3 Q_{yy}}{dt^3} = \frac{G_Nm^2}{3r^2}\ \frac{dr}{dt}}$

All the other components are zero. Substituting this into the Einstein’s formula which I quoted above, the total power radiated comes out to be

${\displaystyle \mbox{Power emitted} = \frac{G_N^3m^2}{15c^5 r^4}\ \left(\frac{dr}{dt}\right)^2 }$

which is terribly small in most cases.

Applications of Classical Physics(Caltech notes) Saturday, May 6 2006

Background: These are some immensely useful notes on many things that are not quantum. It should be useful to everyone.

These are the lecture notes for a course offered by Kip.S.Thorne at Caltech. They cover a wide range of topics in a beautiful way. I am just giving here the chapter names so that you can get an idea.

Chapter 1: Physics in Euclidean Space and Flat Spacetime: Geometric Viewpoint
Chapter 2: Kinetic Theory
Chapter 3: Statistical Mechanics
Chapter 4: Statistical Thermodynamics
Chapter5: Random Processes
Chapter6: Geometric Optics
Chapter 7: Diffraction
Chapter 8: Interference
Chapter 9: Nonlinear Optics
Chapter 10: Elastostatics
Chapter 11: Elastodynamics
Chapter 12: Foundations of Fluid Dynamics
Chapter 13: Vorticity
Chapter 14: Turbulence
Chapter 15: Waves and Rotating Flows
Chapter 15: Waves and Rotating Flows [
Chapter 16: Compressible and Supersonic Flow
Chapter 17: Convection
Chapter 18: Magnetohydrodynamics
Chapter 19: Particle Kinetics of Plasma
Chapter 20: Waves in Cold Plasmas: Two-Fluid Formalism
Chapter 21: Kinetic Theory of Warm Plasmas
Chapter 22: Nonlinear Dynamics of Plasmas
Chapter 23: From Special to General Relativity
Chapter 24: Fundamental Concepts of General Relativity
Chapter 25: Stars and Black Holes
Chapter 26: Gravitational Waves and Experimental Tests of General Relativity
Chapter 27: Cosmology
Appendix A: Concept-Based Outline of Book
Appendix B: Some Unifying Concepts

Posted by :
Loganayagam.R,

Rotation and Time Travel Saturday, Apr 1 2006

General Relativity is one of the most beautiful theories in physics. Simply by changing the global concept of inertial frames to a local version, and connecting the curvature of the space-time manifold to the matter and energy distribution, we get a relativistic theory of gravity. The speed of light is inherited from special relativity, and Newton’s Gravitational constant is inherited by matching the low velocity, low field results with Newton’s theory. No new parameters are introduced into this theory making it highly vulnerable to attack by experimental counter-evidence. In spite of this, it has survived reasonably stringent experimental tests over the last 100 years. It is by far the most appealing theory of classical gravity.

Enter Kurt Godel. This fantastic mathematician caused many a great mathematicians to go mad with his famous incompletenss theorem. He has done a fair amount to perturb physicists as well. In 1949, he published a solution of general relativity, which represented a homogenous rotating universe. This universe was a very strange solution indeed. Firstly it was Anti-Machian. Contrary to the Machian philosophy that the inertial frames would be the set of frames in which the “fixed stars” do not rotate, in this solution the inertial frames see the matter of the universe rotating, and in the frame in which the matter is at rest, coriolis forces act on any moving object. Godel showed moreover, that in this solution, time doesn’t care enough about moving forward forever. He showed that you could have rockets propelled appropriately, that would go in a loop and come back to the same point in spacetime. Such a loop is referred to as a closed time loop (CTC). It is time-like because everywhere, the particle is travelling at a speed less than that of light, and yet it returns to the same point in spacetime. By the same token you could have propelled rockets that return to the same point in space before they left!

You may ask why bother, the solution doesn’t sound like our universe anyway … As far as we know, the matter is stationary in our inertial frames. Why is it surprising that such a strange solution has such strange properties. However, that is not the end of the story. A few years later closed time like loops (CTCs) were discovered in other solutions of general relativity, the Kerr solution, the Von Stockum solution, the Raychoudhari-Som solution and many more. One thing common to all these solutions – Rotation. (Besides rotating solutions, the wormhole class of solutions can also have CTCs). And some of these solutions are not far from reality. In particular when a massive star collapses to a blackhole, it would be described by a Kerr solution, because no matter how slowly it is rotates before collapse, it would gain significant angular velocity as it collapses (conservation of angular momentum).

In the 70s, Stephen Hawkings proposed a chronology preservation conjecture to rule out solutions that permitted “time travel”. However, this proposition did not have any physical grounds, as explicit physical paradoxes could not be constructed with these solutions to rule them out. Some people started proposing designs for time machines based on these solutions, primarily the Von Stockum solution for the rotating cylinder and the Kerr solution for the rotating blackhole. However, people still don’t understand the implications of CTCs in these solutions well enough to conclude without doubt that time travel would indeed be possible in such a case.

A few groups of general relativists are still studying these solutions to understand what exactly a CTC means, and whether their presence would imply a contradiction of physical laws. In particular, they are trying to study what would happen if a thermodynamic system were moved along a CTC. Would its entropy keep increasing as it loops around, or would the entropy increase upto a point, and then necessarily decrease, so that the CTC is closed with respect to entropy too, i.e the entropy returns to the same value every time it goes around the loop. This would suggest that the CTC should be interpreted as time itself being cyclic, rather than a phenomenon of time travel in a universe with time propagating forever forward.

Does general relativity permit time travel? No one yet knows for sure. One thing is clear – the meaning of time in general relativity is not clear.

Posted by: Ravishankar S

P.S : A preprint of the first part of our work can be accessed here http://arxiv.org/abs/gr-qc/0611093