Fun facts

This is a test, an attempt to keep track of various interesting information.

In general the format will be sort of random. Maybe later it will get organized.

Galileo, the Tides, and action at a distance

Cardinal Bellarmine had written in 1615 that the Copernican system could not be defended without "a true physical demonstration that the sun does not circle the earth but the earth circles the sun". Galileo considered his theory of the tides to provide the required physical proof of the motion of the earth. This theory was so important to him that he originally intended to entitle his Dialogue on the Two Main World Systems the Dialogue on the Ebb and Flow of the Sea. The reference to tides was removed by order of the Inquisition.

For Galileo, the tides were caused by the sloshing back and forth of water in the seas as a point on the Earth's surface sped up and slowed down because of the Earth's rotation on its axis and revolution around the Sun. He circulated his first account of the tides in 1616, addressed to Cardinal Orsini (Discourse on the Tides). His theory gave the first insight into the importance of the shapes of ocean basins in the size and timing of tides; he correctly accounted, for instance, for the negligible tides halfway along the Adriatic Sea compared to those at the ends. As a general account of the cause of tides, however, his theory was a failure.

If this theory were correct, there would be only one high tide per day. Galileo and his contemporaries were aware of this inadequacy because there are two daily high tides at Venice instead of one, about twelve hours apart. Galileo dismissed this anomaly as the result of several secondary causes including the shape of the sea, its depth, and other factors.

Galileo dismissed the idea, held by his contemporary Johannes Kepler, that the moon caused the tides.

Discourse on the Tides does not include gravitational forces in its theory to explain the Earth's orbit and does not consider the relation between the ocean and cosmic gravitational forces, like that of the moon. Occurring invisibly, gravity was far too mystic for Galileo's consideration. Galileo did end the Discourse on the Tides with reservations that his theory may be incorrect and the hope that further scientific investigation will confirm his proposal.

This stance towards gravitation and the notion of action at a distance without mediators that Newton introduced was common at the time. Descartes' book of 1644 Principia philosophiae* (Principles of philosophy) stated that bodies can act on each other only through contact: a principle that induced people, among them himself, to hypothesize a universal medium as the carrier of interactions such as light and gravity—the aether. Newton was criticized for apparently introducing forces that acted at distance without any medium.

Beyond that, the theory was widely accepted and there was also a controversy between Newton and Hook regarding who was the father of the theory. It seems that Halley, Wren, De Moivre, and the Royal Society had some part to play in the controversy until in the end Clairaut settled the matter.

In addition, there is this paper on the ArXiv that discusses the issue, titled: "The reception of Newton's Principia"

Newton's Principia, when it appeared in 1687, was received with the greatest admiration, not only by the foremost mathematicians and astronomers in Europe, but also by philosophers like Voltaire and Locke and by members of the educated public. In this account I describe some of the controversies that it provoked, and the impact it had during the next century on the development of celestial mechanics, and the theory of gravitation.

* In Principia Philosophiae one can also find the introduction of Newton's first law of motion, while foundational work on dynamics can be found in Galileo's book Dialogo sopra i due massimi sistemi del mondo (Dialogue on the two main world systems).

References:

[1] Wikipedia: Galileo, Kepler and theories of tides in Galileo Galilei

[2] Wikipedia: Discourse on the Tides

[3] Wikipedia: Historical context in Philosophiæ Naturalis Principia Mathematica

[4] Galileo's Big Mistake

[5] "The reception of Newton's Principia", arXiv:1503.06861 [physics.hist-ph]

R^2 gravity and constrains on the α parameter of the theory

We derive for applications to isolated systems - on the scale of the Solar System - the first relativistic terms in the $1/c$ expansion of the space time metric $g_{\mu\nu}$ for metric $f(R)$ gravity theories, where $f$ is assumed to be analytic at $R=0$. For our purpose it suffices to take into account up to quadratic terms in the expansion of $f(R)$, thus we can approximate $f(R) = R + aR^2$ with a positive dimensional parameter $a$. In the non-relativistic limit, we get an additional Yukawa correction with coupling strength $G/3$ and Compton wave length $\sqrt{6a}$ to the Newtonian potential, which is a known result in the literature. As an application, we derive to the same order the correction to the geodetic precession of a gyroscope in a gravitational field and the precession of binary pulsars. The result of the Gravity Probe B experiment yields the limit $a \lesssim 5 \times 10^{11} m^2$, whereas for the pulsar B in the PSR J0737-3039 system we get a bound which is about $10^4$ times larger. On the other hand the E\"ot-Wash experiment provides the best laboratory bound $a \lesssim 10^{-10} m^2$. Although the former bounds from geodesic precession are much larger than the laboratory ones, they are still meaningful in the case some type of chameleon effect is present and thus the effective values could be different at different length scales. [1]

The question now is, does the idea of saving these models with large α with a chameleon effect (that would make sense in the emptiness of the solar system environment) make any sense in the context of the internal structure of compact objects, where densities are large and therefore the mass of the field should also be large and the interaction length small?

[1] On the 1/c Expansion of f(R) Gravity arXiv:1004.2014 [gr-qc] (Phys.Rev.D81:104003,2010)

Buchdahl limit

The Buchdahl limit states that under some general reasonable assumptions, the compactness of a star can never be smaller than 9/4, i.e., $R/M>9/4=2.25$ (for black holes it takes the lowest value which is 2). It now seems that there is a new limit in town. The additional assumption of a sound speed always smaller than the speed of light results in the limit $R/M>2.74997$.

[1] Maximum mass of a barotropic spherical star arXiv:1503.01517 [gr-qc]

About LaTeX in html

There are several options for introducing TeX in an html environment. My favorite is this (replacemath) where by introducing a short script after the body of the document you can have TeX by using the delimiters $$...$$. A similar alternative option is MathJax. Unfortunately, google sites don't allow scripts or give any option of having something like that.

There is also the option of using MathURL which I don't find very practical since I would rather use equations in my normal flow of writing. Finally, there is this sorry excuse of a TeX engine by Google.

Superluminal motion and pair spot production

Now, this is a fun question and application of special relativity. How would a superluminal moving reflection spot would look like to someone observing it, or for that matter how would an object moving at superluminal speed would look like to an observer? The quick answer is that it would appear as two thing appearing out of nowhere and moving in opposite directions.

[1] Superluminal Spot Pair Events in Astronomical Settings: Sweeping Beams arXiv:1412.7581 [astro-ph.IM]

[2] Photonic Booms

[3] What Does a Faster-Than-Light Object Look Like?

Black Hole Mining

What happens when you have a matter distribution that reaches all the way down to a horizon? What happens when you lower ropes all the way down to a Black Hole? The short answer is that you can extract energy from the black hole faster that regular Hawking radiation would.

[1] Tensile Strength and the Mining of Black Holes arXiv:1207.3342 [gr-qc]

[2] Mining Energy from a Black Hole by Strings arXiv:hep-th\0012260 [hep-th]

Diffeomorphism invariance

Diffeomorphism invariance or general covariance is the invariance of physical laws under coordinate transformations of the form , where is an arbitrary vector.

The requirement therefore is that the gravitational action is unchanged under such transformations (with the addition of some technical requirements regarding the behaviour of the transformation and the variations). This can be expressed as:

, where .

Taking into account in the above expression the fact that

,

this leads to having,

,

which implies the contracted Bianchi identity , since the vector is an arbitrary vector. The same calculation can be performed for the matter action under the assumption that matter is minimally coupled to the metric and that the matter fields satisfy their field equations, i.e.,

. Under these assumptions the invariance of the matter action under coordinate transformations, taking into account the definition of the energy momentum tensor,

,

imply the conservation of the energy momentum tensor, i.e., . The interesting thing is that this can be applied to alternative theories of gravity like f(R), where under the same assumptions for the matter action, the existence of a generalised Bianchi identity is implied.

The result is that one can say that for metric theories of gravity of this type, diffeomorphism invariance implies the contracted Bianchi identity and conservation of the energy momentum tensor.

[1] In latex code within html one should replace + with the string %2B (ASCII Encoding Reference).

[2] Wald 1984, General Relativity, Appendix E, "Lagrangian and Hamiltonian formulation of GR".

[3] Poisson 2004, A relativist's toolkit, Chapter 4.1, "Lagrangian formulation".

[4] Sotiriou, Faraoni, 2010, Rev. Mod. Phys. 82:451-497 ( arXiv:0805.1726 [gr-qc])

A rolling coin

Inspired by this year's panhellenic physics exams, here are some animations of rolling coins in the interior and exterior of a circular rim.

How many turns does the coin make? ;)