# Category Archives: gravity

## Quantization of event horizon radius and Quasar Redshifts

I’m getting ready for the FFP10 meeting later this month. In reading the abstracts of those who will be giving talks or posters, I came upon “Analyses of the 2dF deep field” by Chris Fulton, Halton Arp and John G. Hartnett. The abstract is about the relationship between low redshift and high redshift astronomical objects. The claim is that some quasars have redshifts that do not give their true distance; instead, they are much closer. Looking on arXiv finds: The 2dF Redshift Survey II: UGC 8584 – Redshift Periodicity and Rings by Arp and Fulton.

If these high and low redshift objects actually are related, this places doubt on the Hubble relation. In addition, when low and high redshift objects appear to be related, their redshifts are related by quantum values . From observations, Arp has proposed that quasars evolve from high to low redshift, and finally become regular galaxies.

Now for quasars to have redshifts that differ from their true distances implies that their redshifts are determined gravitationally; that is, what we are seeing is partly the redshift of light climbing out of a gravitational potential. And if these redshifts are quantized, this gives a clue that the structure inside the event horizon of a black hole is not a simple central singularity but instead there must be repetitive structure.

In a classical black hole, the region inside the event horizon can only be temporarily visited by regular matter. Even light cannot be directed so as to increase its radius in this region. Let’s refer to this region as the “forbidden region” of the black hole as it is near the central singularity. For the classical black hole, this includes everything inside the event horizon. We will be considering the possibility that the forbidden regions of a black hole occur as infinitesimally thin shells, and that between these shells, light can still propagate outwards:

Forbidden regions shown in red.

Event Horizons as Quantum amplitudes

If we were looking for a quantum mechanical definition of the inside of a black hole, we could define it as the region where particles have a zero probability of moving outwards. We could say that the transition probability for the particle moving outwards is zero. However, in quantum mechanics probabilities are defined as the squared magnitudes of complex amplitudes. The way we compute transition probabilities is from complex transition amplitudes. If the transition amplitude between two states is zero, we say that they are “orthogonal”. Zero transition amplitudes correspond to points where a sine wave is zero; at these points, deviations to either side give nonzero transition amplitudes:

How to get zero probabilities from nonzero in QM.

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## My Gravity paper accepted for publication

I’ve just got notice that my gravity paper, titled The force of gravity in Schwarzschild and Gullstrand-Painleve coordinates has been accepted for publication in the International Journal of Modern Physics D, with only a very minor modification.

I’m kind of surprised by this, given that the paper proposes a new theory of gravity. I was expecting to have that portion excised.

And to help make a week more perfect, my paper for Foundations of Physics, titled Spin Path Integrals and Generations, got a good review along with a nasty one (and much good advice from both), and the editor has asked for me to revise the manuscript and resubmit. So I suppose this paper will also eventually be published. I’m a little over half finished with the rewrite. This paper is, if anything, even more radical than the gravity paper.

Finally, the Frontiers of Fundamental and Computational Physics conference organizers have chosen my abstract (based on the Foundations of Physics paper) for a 15 minute talk. The title is Position, Momentum, and the Standard Model Fermions. Marni Sheppeard (my coauthor for a third paper, “The discrete Fourier transform and the particle mixing matrices” which so far is having some difficulty getting published), is giving a related talk, Ternary logic in lepton mass quantum numbers immediately following mine.

So all in all, I am a very lucky amateur physicist

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## Dark Flow, the Speed of Gravity, and the CMB

Kashlinsky, Atrio-Barandela, Kocevski, and Ebeling have just put out a preprint on the peculiar motions of galactic clusters: A measurement of large-scale peculiar velocities of clusters of galaxies: results and cosmological implications. In short, they claim that all galactic clusters appear to have a motion with respect to the cosmic microwave background (CMB). The motion of a galactic cluster slightly effects the energy of the microwave radiation that travels through it, so they use the temperature map of the CMB to determine the velocity of those galactic clusters.

And the result is that the whole (observable) universe appears to be moving with respect to the CMB. This was not expected because the observable universe is approximately isotropic and so shouldn’t be going anywhere. They write (in the abstract):

This flow is difficult to explain by gravitational evolution within the framework of the concordance LCDM model and may be indicative of the tilt exerted across the entire current horizon by far-away pre-inflationary inhomogeneities.

However, the tilt is easy to explain when you assume that the speed of gravity is larger than C: If gravitational interactions travel faster than light, you will automatically be able to feel the gravitational attraction of matter even if it is too far away for you to see.
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## Lorentz Violation and Feynman’s Checkerboard Model

Lubos Motl brings to our attention a paper by Ted Jacobson and Aron C. Wall on black hole theremodynamics and Lorentz invariance, hep-ph/0804.2720 and claims that theories that violate Lorentz invariance are ruled out because they will also violate the second law of thermodynamics, the law that requires that entropy never decreases. Lubos concludes, “At any rate, this is another example showing that the “anything goes” approach does not apply to quantum gravity and if someone rapes some basic principles such as the Lorentz symmetry or any other law that is implied by string theory, she will likely end up not only with an uninteresting, ugly, and umotivated theory but with an inconsistent theory.” I disagree with this.

First, the abstract of the article:

Recent developments point to a breakdown in the generalized second law of thermodynamics for theories with Lorentz symmetry violation. It appears possible to construct a perpetual motion machine of the second kind in such theories, using a black hole to catalyze the conversion of heat to work. Here we describe the arguments leading to that conclusion. We suggest the implication that Lorentz symmetry should be viewed as an emergent property of the macroscopic world, required by the second law of black hole thermodynamics.

From the abstract, we see that Lubos has put the cart in front of the horse. Rather than proving that Lorentz symmetry has to be exact “all the way down”, the authors instead say that Lorentz symmetry does not have to be present at the foundations of elementary particles because it will automatically emerge macroscopically as a result of requiring that the second law of thermodynamics apply to black holes. And I agree wholeheartedly with this.
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## The Painleve Equations of Motion

In the general theory of relativity, the orbits are given by geodesics. A geodesic is a path that extremizes the path length. The path length is defined as the integral of $ds$ over the path, where $ds^2$ is the metric. For the case of Painleve coordinates on the Schwarzschild metric, $ds^2$ is given by:
. Let’s let our path start at time t=0 and end at time t=1. For the path to be a geodesic, we must extremize the following integral (I’ll quickly sneak in a minus sign to make the path be timelike instead of spacelike):

To make life easier for us, we will make the assumption that the orbital motion is in the $\theta = \pi/2$ plane so there’s no $\theta$ dependence. That turns the angular part of the square root into $r^2\;(d\phi/dt)^2$. Furthermore, since the simulation is going to use Cartesian, (x,y) coordinates, we might as well replace $r^2\;(d\phi/dt)^2$ with $(x\;dy/dt-y\;dx/dt)^2$, and $dr/dt$ with $x\;dx/dt + y\;dy/dt$, their Cartesian equivalents. And put M=1, we can always fix it later by dimensional analysis.
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## Painleve Coordinates

In the previous post, we took a tour through the literature and found that when general relativity is translated into the elegant mathematical language of the geometric algebra, the natural coordinates for a black hole turn out to be Painleve or Gullstrand-Painleve coordinates instead of the more common Schwarzschild coordinates. Our next post will derive the equations of motion for orbits in this coordinate system, but before we get into the difficult mathematics, we should take a quick look at the Painleve coordinates.

First of all, most of my readers will know that in general relativity, the choice of coordinates is quite arbitrary. Both Schwarzschild and Painleve coordinates describe the same object, the gravity field of a gravitating object which is spherically symmetric (and therefore non rotating), i.e. they are all descriptions of the black hole. In this sense they correspond to the same solution to Einstein’s field equations, which is sometimes called “Schwarzschild’s Solution”, or the “Schwarzschild Metric”. This is a little confusing, “Schwarzschild” was the person who found the Schwarzschild metric and he found it using the Schwarzschild coordinates, so his name is used twice here.

I guess I should put a pretty picture from the gravity simulation that resulted from all this here so it will show above the fold. This is a set of “knife edge” orbits, that is, orbits that quite nearly fall into the black hole but do not. Due to the time spent near the black hole, whose event horizon is marked in gray, the test masses get huge precession:

We will be discussing the less pretty, but more mathematical subject of Painleve and Schwarzschild coordinates in this post.

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## General Relativity, Painleve and QFT

The first problem in writing gravitation as a particle interaction is the fact that QFT works best on flat space, while general relativity is almost always written in arbitrary coordinates. One of the underlying principles of general relativity is that coordinates shouldn’t matter (background independence), so this problem appears to be a deep one. The point of view we will take here is that of the “new physics” .

That is, we will treat the equations of the old physics with more respect than we treat their theories. Consequently, instead of chasing after will-o-the-wisps like background independence, we will instead search for a method of writing general relativity using the mathematical tools of quantum field theory. Very fortunately for us, that method has already been found; it is the gauge theory of gravity discovered by the Cambridge Geometry Algebra Research Group. The purpose of this post is to introduce the theory to those who have not yet been exposed to it, and to note that this gravity theory (which is identical to GR so long as you restrict your attention to stuff that happens outside of the event horizons of black holes) picks out Painleve coordinates as a natural flat space (and therefore QFT compatible) coordinate system for a non rotating black hole.

Those with a graduate education in physics are already familiar with the Geometric Algebra (GA) in that it is equivalent to the Gamma matrices used throughout quantum field theory. So a gravitation theory that is equivalent to general relativity, but is written with gamma matrices, is a natural starting point for a search for a unified field theory.

The primary proponent for the use of GA in physics (outside of QFT) is David Hestenes, who applied it to classical and quantum mechanics. As the introduction to GA article at the Cambridge Geometry group’s website puts it:

We believe that there are two aspects of Hestenes’ work which physicists should take particularly seriously. The first is that the geometric algebra of spacetime is the best available mathematical tool for theoretical physics, classical or quantum. Related to this part of the programme is the claim that complex numbers arising in physical applications usually have a natural geometric interpretation that is hidden in conventional formulations. David’s second major idea is that the Dirac theory of the electron contains important geometric information, which is disguised in the conventional matrix based approaches.

Now that’s a pretty big claim: that geometric algebra is the best mathematical tool for all physics. I will spend the rest of this post exploring this claim in the case of general relativity, and then tracing the consequences for a unified field theory.
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## Kepler’s Symmetries and Newton’s DE

This is the first of a series of posts about gravity, and the java applet I wrote that simulates Newtonian and relativistic orbits around a (non rotating) black hole. We begin by discussing the relationship between Kepler’s observations and Newton’s universal law of gravitation.

Usually when you write a scientific applet, most of the effort is in the java coding rather than in finding the equations. That is very much true about Newton’s equations of motion around a black hole, which are very easy. His Law of universal gravitation is:

where F is the force, G is the universal constant, M is the mass of the black hole (or other spherically symmetric gravitating body), and m is the mass attracted. We will simplify this. We will use units with GM = 1, let m << M, rewrite Force as mass x acceleration, choose Cartesian coordinates (x,y,z) so that $r^2 = x^2 + y^2 + z^2$, only consider motion in 2 dimensions so z=0. The resulting simplified differential equation (DE) is:

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## Measuring the Speed of Gravity (Waves)

Newton’s equations give the speed of gravity as infinite. For example, in Cartesian coordinates, suppose a gravitating mass 2M is at the origin up until time t=0.  At that time, the mass splits into two masses of mass M, one going in the +x direction at speed v the other in the -x direction at speed v. For times greater than 0, the gravitational potential is given by the sum of the two gravitational potentials:

(1) $\;\Phi(x,y,z,t) = \frac{GM}{\sqrt{x^2+y^2+(z-vt)^2}} + \frac{GM}{\sqrt{x^2+y^2+(z+vt)^2}}.$

At any distance, the above depends on t so the gravitational potential (and it is easy to show the gravitational force) is instantaneously changed at all distances from the origin. The speed of gravity is therefore infinite in Newton’s theory.
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