Monthly Archives: February 2012

General Mathematics

The Alan Turing Year-2012

Leave a comment

Just a reminder that this year is The Alan Turing Year. Turing was born in London on the 23th June 1912.

There is just about no area of science that Turing has not had some impact on in some way. I think he is best known for his pioneering works in computer science.

In a wider context Turing is famous for his code breaking work during the second word war at Bletchley Park. After the war he worked at the National Physics Laboratory creating the designs for ACE, which was a very early electronic stored-program computer.

In 1948 Turing joined the Computing Laboratory at Manchester.

Turing’s story after that is quite sad. He was prosecuted for homosexual acts in 1952, he was given a chemical castration after that. He died of cyanide poisoning on 1954. The verdict of the inquiry was suicide.

You can find out lots more about Turing’s influence on mathematics and science at the official The Alan Turing 2012 homepage.

Other Links

Wikipedia

Andrew Hodges’s page

The MacTutor biography

Book Reviews

Space, time, and gravity: the theory of the big bang and black holes, by Wald

Leave a comment

The general theory of relativity has a reputation of being very difficult to comprehend. This is especially true for the layperson or undergraduate students. Space, time, and gravity: the theory of the big bang and black holes by Robert M. Wald plots a very accessible course through the core phenomena of general relativity without any unnecessary mathematical detail.

The book is based on lectures that Wald gave in Chicago 1977. The 1992 version is updated due to the great advancements in observation cosmology since the first edition.

The book consists of 10 main chapters, an appendix (in the 1992 edition) and a list of suggestions for further reading.

Chapter 1 describes the geometry of space and time from a Newtonian “everyday” perspective. The ideas of spce-time diagrams, the principle of relativity and absolute simultaneity are given here. In the next chapter it shown how these ideas need modifying following Einstein.

Chapter 2 introduces special relativity. The principle concepts of Einstein’s axioms, relative simultaneity, the space-time interval, Lorentz contraction and time dilation are neatly presented. Wald emphasises the geometric way of thing which is paraphrased as “it is all in the metric”. With this in mind, geodesics in Minkowski space-time are discussed and the conclusion is drawn in special relativity the space-time is flat .

General relativity is the subject of Chapter 3. Wald discusses why Newtonian gravity cannot be directly incorporated in to special relativity. The key trouble is the notion of relative simultaneity and instantaneous interactions. The solution is that equivalence principle strongly suggest that space-time is not flat. Wald then moves on to discuss the Einstein field equations, the bending of light, and the gravitational red shift.

Chapter 4 discusses the implications of general relativity for cosmology. Principally it is argued that the Universe must have started from a singular point and that the “Big Bang” is rather unavoidable in the the context of classical general relativity. Wald describes the large scale structure of the Universe, the “Big Bang” and Hubble’s law. In the final part he discusses singularity theorems and questions if we should take singularities seriously? Only quantum gravity will really shed light on this…

Chapter 5 presents a walk through the evolution of the Universe. Evidence for why we think the Big Bang is a good theory is presented here. For example the CMBR and the cosmic abundance of deuterium.

Stella evolution up to white dwarfs and neutron stars is the subject of Chapter 6. Wald discusses stellar birth, the continued evolution of stars, teh electron degeneracy pressure and white dwarfs, supernovae and neutron stars.

The gravitational collapse to form black holes is the topic of Chapter 7. The basic structure of black holes is presented as is the cosmic censorship hypothesis and the no-hair theorem. Due to the singularity theorems of Penrose and Hawking, and other works it is generally accepted that complete gravitational collapse will always result in a black hole.

Chapter 8 discusses the Penrose process of energy extraction from a rotating black hole. Wald discusses the notion of ennergy-momentum in special and general relativity. The complications of what we mean by energy in general relativity are highlighted. The Ergosphere of a rotating black hole is described as is the Penrose process and the energy extraction limits.

The astrophysics of black holes is the subject of Chapter 9. Wald discusses the proposed mechanisms of black hole production: collapse of a massive enough star, collapse of a cluster of stars and primordial black holes. Wald then discusses gravitational lensing, gravitational radiation and X-ray sources.

Chapter 10 discusses particle creation near a black hole and Hawking radiation. The consequence for conservation laws of black hole evaporation are discussed as is the strong link with thermodynamics.

We have learnt a lot since the 1977 edition and also the 1992 edition about observation cosmology and the deep link between black holes and thermodynamics. That said, the book is a wonderful expose to these deep ideas in a way that is rather accessible.

As a note, I own the first edition from 1977 and so based the above review on that. From what I can tell the second edition is very similar but the numbers are updated.

Paperback: 164 pages
Publisher: University of Chicago Press; 2nd Revised edition edition (1 Mar 1992)
Language English
ISBN-10: 0226870294
ISBN-13: 978-0226870298

General

Metaphysics: not science

7 Comments

Metaphysics is a dark ocean without shores or lighthouse, strewn with many a philosophic wreck.

Immanuel Kant

Metaphysics is really the branch of philosophy that contemplates the questions of existence, being, the origin of the Universe and similar questions. Unfortunately, the term has also been perverted to mean spiritualism, magic and “experiences beyond physics”.

Let us look at the dictionary definition:

  1. the branch of philosophy that treats of first principles, includes ontology and cosmology, and is intimately connected with epistemology.
  2. philosophy, especially in its more abstruse branches.
  3. the underlying theoretical principles of a subject or field of inquiry.

which we adapt from Dictionary.com

Why it is not science

On the face of it metaphysics seem to be very similar to science. Both subjects want to understand the natural world around us.

The big difference is that science is based on empirical evidence. Science is about exploring the world around us and putting our theories to the test by making empirical predictions. A theory is only scientific if it, at least in principle, makes predictions that we can test.

This is very closely related to the scientific method, which serves as a guideline to scientific thinking. Simplified the scientific method is

  1. Observation: use your experience of the world. Consider some phenomena.
  2. Theory: make some mathematical theory that explains the said phenomena.
  3. Prediction: use your mathematical theory to make predictions beyond the initial phenomena.
  4. Test: you now look for the predicted phenomena. If you don’t find it you go back to step 2.

The above is of course over simplified and idealistic. The point is one has to make clear predictions that can be tested.

Metaphysics fails here

Metaphysics is not constrained in this way. Metaphysical ideas cannot usually be put to the test via empirical evidence. This means they cannot be falsified. Importantly this means that differing positions in metaphysics cannot be supported or refuted based on experimental evidence.

Therefore, metaphysics requires some belief. You can argue a metaphysical position based on your opinion and maybe some philosophical consequences of this position. However, you could never appeal to experimental or observational evidence. If you could, it would be science!

The lesson for us all

So, when people make claims that they have a theory of everything or a theory of the atom based on high school mathematics or any thing similar you must ask yourself “is it science?”

By this I mean they should have a mathematical framework in which one can make calculations of physical phenomena that can, at least in principle be tested.

If this is not the case then at best it is metaphysics, at worse pseudoscience.

When he to whom one speaks does not understand, and he who speaks himself does not understand, that is metaphysics.

Voltaire

Physics

Professor Higgs receives award

4 Comments

Professor Peter Higgs has been awarded the the Edinburgh Award 2011. The award is organised by the City of Edinburgh Council. The recipients are nominated by the citizens of Edinburgh for their outstanding contribution to the city.

It was back in 1964 that Higgs developed the theory of spontaneous symmetry breaking in the context of particle physics. The Higgs boson is an important, but as of yet undiscovered component of the standard model of particle physics. CERN in 2011 narrowed down the possible mass of the Higgs boson and it is hoped that a proper discovery will be made this year.

The Edinburgh University news report can be found here.

Professor Higgs website can be found here.

The BBC news report can be found here.

Physics

Superluminal neutrinos, or a faulty cable?

Leave a comment
flasks

We all remember the hype and controversy that followed after the OPERA collaboration at CERN released their report saying that they measured neutrinos travelling faster than the speed of light. The report can be found here.

The OPERA collaboration how identified a rather mundane possible solutions to this…

CERN has released a statement to confirm that the OPERA collaboration has identified a feature of its experiment that could explain its puzzling superluminal-neutrino discovery – a faulty optical fibre. The collaboration is now investigating this, and one other potential source of error, and it plans to carry out new experimental runs in May.

James Dacey is multimedia projects editor for Physics World

There now appears two potential faults with the experiment. First is a faulty optical cable which transmits data in the experiment. Second there maybe issues with the way the GPS systems were synchronised giving an over estimation of the distance travelled by the neutrinos.

Looks like Jim Al-Khalili will not have to eat his boxer short live on TV. Unless he wants to that is!

See the Physics World report here.

See the CERN news report here.

Physics

UK research in physics now second in the world

2 Comments

The UK has overtaken the US in terms of the quality of physics-research output, according to a new report carried out by Evidence, which is owned by information-services provider Thomson Reuters. The report, Bibliometric evaluation and international benchmarking of the UK’s physics research, states that the UK is now second to Canada when ranked on the quality of research papers, measured as the average number of times that such papers are cited.

Michael Banks, news editor of Physics World.

For a relatively small country the United Kingdom of Great Britain and Northern Ireland (to use the full name) does rather well in the impact of science research.

 

You can read the full Physics World news report  here.

General Mathematics

Felix Alexandrovich Berezin: the first supermathematician

Leave a comment
Berezin (25 April 1931 – 14 July 1980) was without a doubt the first person to really work with supermathematics, that is mathematics with commuting and anticommuting variables. His worked paved the way for mathematicians and theoretical physicists.

Berezin also made many important contributions to quantisation, he is best known for the notion of integration with respect to Grassmann variables and the generalisation of the determinant to a “super-determinant” known as the Berezinian.

Alexander Karabegov, Yuri Neretin and Theodore Voronov [1] have written a survey of Berezin’s work which has appeared on the arXiv.

Their survey concentrates on Berezin’s contributions to representation theory, the general concept of quantization, and supermathematics.

References

[1] Alexander Karabegov, Yuri Neretin, Theodore Voronov. Felix Alexandrovich Berezin and his work. arXiv:1202.3930v1 [math.HO].

Physics

Energy is not frame independent

5 Comments

For a closed system we have energy conservation and this is deeply tied into the notion of the physics being unaltered under shifts in time. A salient point here is that we have fixed a frame in which to measure the energy. It is not true that the energy is the same measured in any inertial frame, what is true is that it is conserved.

A simple example

Let us consider a free particle of mass \(m\) in one dimension. Let us pick some coordinates \((x,t)\). With respect to this frame we have that the kinetic energy

\(E = \frac{1}{2 m}p^{2}\),

where the momentum \(p \) is given by \(m v = m \dot{x} \). The dot is the derivative with respect to time. (I am using the usuall abuses of notaion here, this should not confuse)

Now let us consider a Galilean transformation

\(x’ = x {-} ut\)
\(t’ =t\).

Then we calculate how the momentum transforms

\(p = m \frac{d}{dt}\left( x’ – ut \right) = m \left( v’ +u \right)= p’ + mu\).

Or \(p’ = p – mu\).

This is exactly what you would expect. So on to the energy…

\(E = \frac{1}{2m}p^{2} = \frac{1}{2} \left(p’ + mu \right)^{2}\)
\(= E’ + \frac{1}{2}mu^{2} + up’ = E’ – \frac{1}{2}mu^{2}+ up\)

so

\(E’ = E + \frac{1}{2}mu^{2} -up\).

Clearly \(E’ \neq E\)

.

Recap

Energy conservation should not be confused with energy being observer independent, it clearly is not.