Gravitational Lensing for the Masses: Delivered

   So I gave the lensing talk tonight, I think it went reasonably well.  My professor interrupted me once to check to see if there were any questions from the group; some people said I was going a bit fast, so he told me to slow down.  But then he said: remember, this is a review talk, and is supposed to be helpful mainly for the first year graduate students, who may have not seen gravitational lensing in their classes; however, don’t forget you also have a world-renowned expert on gravitational lensing in your audience (a visiting professor).  So I slowed down for the benefit of the first years, which would have been fine, except that there was only one other first year at the meeting, and she heard me practice the talk the night before.  All of the other first years were off somewhere finishing homework.  So, I probably bored the world-renowned expert to death.  I guess it’s ok, since he got free pizza.

   On a side note, I read an article yesterday about how the word “so” is to scientists as “um” is to everyone else.  Re-reading the paragraph I just wrote, and my previous posts…man.  Hopefully I didn’t throw out too many sos (soes?  so’s?) in my talk, although the article seems to think that sos are good.  If you say so, article.

   I’ll try to update the post below with the rest of my talk at some point in the future.  Yeah…probably not gonna happen any time soon.

Gravitational Lensing for the Masses

   Spring quarter has just started up; I am planning on taking it easy this term, enrolling only in the two required astronomy courses and one physics course.  Even so, I will probably be busy enough to continue to neglect writing here more often.  So, I will try to crank out at least this post before things pick up.

   Our theory group has been devoting half of each of our weekly meetings to reviewing classic papers.  My turn to present is at the end of the month, but I’d like to start throwing something together – hence, this post.  I’ll keep adding things in bits and pieces as I go.

   The paper I chose to review is a classic by Blandford and Narayan: Fermat’s Principle, Caustics, and the Classification of Gravitational Lens Images.  I have a fond spot for this paper, since it is the first paper I was asked to read when I began doing research.  I was a naïve freshman back then, and I emailed pretty much every professor in the physics department looking for someone who would give me a project.  I got a surprising number of replies, mostly saying things like “We don’t have any open projects, but would be glad to help you find one.”  I received two positive responses: one from a professor in the astrophysics department, asking me if I was interested in working on gravitational lenses, and another from a professor in the theoretical physics department who researched string theory (again, I was a naïve freshman; I thought I was only years away from finding a grand unified theory just because I read Brian Greene’s The Elegant Universe). 

   I met with the string theory professor, and he asked me to read this paper to see if I could write some code to replicate and improve on the results.  It’s been four years and I still have no idea what the hell the paper is talking about.  The astrophysics professor asked me to read Blandford and Narayan, and I ended up working with him for the rest of my undergraduate years.

   So basically, the important results of this paper should be accessible to anybody who is smarter than I was when I was a lowly freshman, which should be nearly everybody.  All you will need to know is some simple multivariable calculus, and be willing to take one or two results from general relativity on faith; then you should be able to understand Blandford and Narayan’s elegant presentation of gravitational lensing.

Gravitational Deflection of Light by a Point Mass

   One of the first important tests of general relativity was confirming that the deflection of light by mass was predicted correctly by the theory.  This was confirmed (albeit with extremely large error bars) by the Eddington expedition, which measured the angle of deflection of starlight passing by the sun; such an observation could only be made during a solar eclipse.  Subsequent observations have verified that indeed the angle of deflection predicted by general relativity (exactly twice the angle predicted by a Newtonian calculation) is observed.

   So what is the expression for the angle of deflection (which we will denote by )?  Despite the importance of this calculation in the history of general relativity, we can actually find the answer quite simply using dimensional analysis.  Consider the figure above (taken from here).  The light passes by the star with a given impact parameter, or distance of closest approach of the light ray, which is denoted by (not exactly the distance of closest approach, as drawn in the figure, but since the angles involved are small the figure should suffice to get the idea across).  Clearly, this impact parameter  should enter into the expression for the deflection angle ; the closer the light ray passes, the more it should be deflected.  The other important physical variables in the problem are the mass of the deflecting object (the smaller the mass, the smaller the deflection angle), strength of the gravitational constant (the weaker the strength of gravity, the smaller the deflection angle), and the speed of light (the faster light goes, the less it should be deflected).

   We now have four physical variables , , , and , which we must somehow combine to form a dimensionless quantity .  By using dimensional analysis (more exactly, the Buckingham Theorem), we see that the only way this can be done is: .  That is, the deflection angle is some function of the dimensionless quantity .  We can easily see this quantity is dimensionless; has units of (gravitational potential) energy/mass, and also has units of (kinetic) energy/mass.  Physical intuition (along with our arguments above on how changing each variable should affect the size of the angle) tells us that we should have , and that the deflection angle is thus given by:

     .

   We have found roughly the right answer just from doing dimensional analysis!  The Newtonian calculation simply gives a correction ; this results from integrating the deflecting component of the gravitational force along the light ray, i.e.

    

Using , we then have:

    

   To find the exact answer using general relativity, the calculation is a little more involved.  We still must add up the deflections using an integral along the trajectory; however, now we cannot simply write for the gravitational force.  Instead, we must use the geodesic equation , using the Christoffel symbols of the Schwarzschild metric in the weak-field to calculate the covariant derivative.  This gives relations between the ; the relation of interest is (assuming ).  Thus, we have:

     .

Thus, the general relativistic calculation simply gives , an extra factor of two greater than the Newtonian result.  Finally, we have found that the deflection angle given by general relativity is .

Cosmology? That’s like, makeup?

   I am back at graduate school after visiting home for the holidays.  While it was nice to catch up with people, it was a little depressing to realize that almost nobody has any idea of what I actually do.  Whether it is old friends from high school, my family, people that talked to me on the plane, even my roommates here at graduate school; most people are unsure, at some level, of what it is exactly that theoretical astrophysicists do.  I find that there are mainly two levels of confusion:

   I)  Quite confused.  When I say I am studying astrophysics, these people say, cool!  So, you want to design airplanes?  No?  Then, spaceships!  (This is by far the most common response.  But once, when I told someone sitting next to me on a plane that I am studying cosmology…see title.)  These are not uneducated people, either; most of them are very smart, with college educations, and all of them are capable of understanding what we do (at least, that is what I like to believe).  So at some level, the methods used to communicate even the basics of our science to the public – popularization through books or news articles, museums, public lectures, etc. - are very broken.

   II) Just a little confused.  These people know enough to ask me, so, what is up with the big bang?  Have you found that Higgs boson yet?  Or, tell me about string theory!  But, they still hold misconceptions about how science works.  They’ll say, oh, cosmology is nice, but in the end, you can’t really prove anything, right?  It is almost like philosophy.  I have heard this view many times from many people.  Apparently, cosmologists sit at a desk, smoke pipes, and make up stories about how they think the universe began.  Whoever thinks up the nicest story wins.  Then they will write up fancy articles – e.g., “A Discourse on the Cosmological Theory of Turtles All the Way Down” – that contain nice ideas, but are just creation myths in the end.  So you see, cosmology is just another one of the liberal arts.

   So, what’s wrong with this picture?  I think that Type I is mostly a failure of our public education system.  Physics (and other sciences) as taught at the public high school level is a joke at best and a tragedy at worst.  Mostly, high school physics fools people into thinking physics is difficult and consists only of inclined planes and pulleys, while teaching them nothing that is actually challenging or inspiring at all.  In college, physics is rarely required; by this time, it is usually too late anyways, and more courses just end up making the problem worse.  Attempts to fix the problem with science popularizations and news articles are futile, because nobody will read more about something they have already been taught to hate.  These symptoms manifest even at places such as my alma mater, which is supposed to be a bastion of science education.  The Type IIs are those whose interest in science was strong enough to survive this system, but for whatever reason hasn’t been further cultivated.  Here, the failure is more on a personal level.  Maybe the popularizations and the articles, which Type IIs actually do read, aren’t clear or engaging enough; or in the same vein, maybe the scientists which they actually know – i.e., me – just aren’t that good at communicating the essentials of their field.  I encourage these people to keep reading and keep asking questions.

   So, what do I do, in a nutshell?  Let me start with physics.  Simply put, this is just understanding how the world works.  Once you understand a little bit, you begin to see that you can and must do this in a mathematical way; you could just wax philosophical, as the Greeks did, but our understanding has progressed enormously since then.  Once you have a mathematical theory, you keep checking it against experiment.  If you find an experiment that cannot be predicted or explained by your theory, then you modify your theory.   And so on.  So our understanding of the physical laws of the universe walks forward step-by-step on these two feet, theory and observation - what we think and what we see.  It is just a bonus that our theories turn out to be simple (relatively) and beautiful (absolutely); that’s just how the universe works.

   So then, what is astrophysics?  This is just, literally, physics of the stars.  It is a wonderful fact that, so far as we know, physics works the same everywhere.  So we can apply what we learn here in our tiny little corner of the universe to everything else we see.  We can use the Newtonian theory of gravity we deduce from falling apples to understand the orbits of the planets.  Likewise, we can use observations of astronomical events that we don’t understand to correct our theories.  So, when we see that the orbit of Mercury isn’t predicted correctly by our Newtonian theory of gravity, we must construct a theory of general relativity that explains the problem.  Such principles hold not only in our solar system, but beyond.  Newtonian gravity explains the orbits of binary stars, but we must turn to general relativity to explain the orbits of binary neutron pulsars, and so on.

   And so cosmology is just application of theory and observation to even larger scales – not just stars, but galaxies of stars, and the entire universe of galaxies.  Far from being just-so stories, our theories of how the universe began are based on observation.  If we observe that a nearly uniform background radiation permeates the universe and that other galaxies are moving away from us, we theorize that the universe began in a hot explosion and has been cooling and expanding since then.  We use our theory of general relativity to run the clock backwards to find what the initial temperature must have been, and using our theories of nuclear physics calculate what the ratio of the different elements must have been at that temperature.  We then run the clock forward again, and check to see if this ratio is what is observed in our current universe.  It is.  So we have done science.  We are not just making up fanciful stories; we are trying to understand how the universe began as precisely and quantitatively as we can.

   To improve on this cosmology, by considering observational puzzles such anisotropies in the background radiation, the distribution of galaxies and matter in the universe, dark matter and dark energy, and many more - that is why I am going to graduate school.  Hopefully, by understanding these puzzles, we can improve our physical theories.  I’m drawn to cosmology because the universe is such a fantastic experiment – the only one that allows us to probe such gigantic distances and high energies.  (It is often said that to test that our theories hold at the enormous energies present during the big bang, we’d need a particle accelerator the size of our solar system.  So, particle physics won’t do it for me.)  This is why I do cosmology – to increase our understanding of the physical laws of the universe.  It has nothing to do with makeup, or making up stories!

   I am reminded of the dedication from the general relativity textbook by Misner, Thorne, and Wheeler (don’t tell anybody, but I like the dedication more than I like the book itself):

We dedicate this book
To our fellow citizens
Who, for love of truth,
Take from their own wants
By taxes and gifts,
And now and then send forth
One of themselves
As a dedicated servant,
To forward the search
Into the mysteries and marvelous simplicities
Of this strange and beautiful Universe,
Our home.

   Unfortunately, sometimes the servants forget to report back, or find that no words can adequately describe what they have found. Sometimes, you just have to see it for yourself.

Differential Geometry and Maxwell’s Equations I: Manifolds and Vector Fields

   I have been rereading the first part of Baez and Muniain, on reformulating electromagnetism in the language of differential geometry.  Here are some notes; they mostly follow the book, but only the parts necessary for writing down and understanding the final equations.

Manifolds

   The spaces physicists are interested in studying are locally similar to .  For example, the 2-sphere – i.e., the surface - is locally similar to the plane , which is why the world looks flat.  To generalize this idea to spaces that look like , we define n-dimensional manifolds.

   First, we must define topological spaces.  A topological space is a set and a collection of subsets of , called the open sets, which satisfy: 1) the null set and are open, 2) the intersection of any two open sets is also open, and 3) the union of any number of open sets is also open.

   Topological spaces allow the definition of continuous functions.  A function between two topological spaces, , is continuous if given any open set in the target space, the inverse image  is open.  Basically, this means the function sends “nearby” elements of  - “nearby” meaning the elements are all members of some open set – to “nearby” elements of .

   Now, we can see how to use the idea of open sets to link topological spaces and .  Consider a topological space , covered with open sets .  For each open set, we define a continuous function, called a chart: .  If these charts are defined such that the transition function  between the two spaces associated with the open sets  and is smooth, then is called a smooth n-dimensional manifold.  This manifold is a space of elements, separated into “nearby” groups, each of which can be related to ; thus, the manifold looks like smoothly-connected patches of n-dimensional Euclidean space.  Using these local Euclidean spaces, we can define smooth functions on the manifold in a familar manner; e.g., is smooth if is smooth for all .  We can then use these functions to define vector fields, and more, on the manifold, as if we were working in as usual.

Vector Fields

   We are used to thinking of a vector  as a coordinate n-tuple, with n components : .  We multiply these components by the “basis vectors” and add everything up to get the vector “object.”  However, this picture is unclear for manifolds; how should one define simultaneous basis vectors for two vectors at different points on the sphere, for example?  If one thinks of these two vectors as little arrows tangent to the sphere, they might lie in different planes entirely, which makes defining basis vectors and coordinates difficult.  Thus, we would like a coordinate-free definition of vector fields.

   To start, we note that one can differentiate a function on the manifold in the direction of a vector, with the directional derivative.  Consider the usual directional derivative of in the direction of , which we shall write as ; this is simply .  Let us identify the vector field not with the components , but instead with the operator .  What does this mean?  Given a vector field and a function on the manifold, at each “point” of the manifold we’ll take the derivative of in the direction of , giving us a new function .  So combining a vector field with a function gives us another function on the manifold, which is related to derivatives of the original function.

   Now we can generalize this new picture of a vector field to a coordinate-free abstraction that contains all of the essential properties of the directional derivative.  Let be the set of real-valued smooth functions (which have infinitely many continuous derivatives) on the manifold .   is a commutative algebra over , which just means that elements of combined with elements of  obey a number of addition, multiplication, and distributive properties.  We define a vector field on to be a function , obeying: 1) linearity over , and 2) the Liebniz law , where .  Let be the set of all vector fields on .  It is easy to check that is a vector space over .  Thus, this abstract definition satsifies our usual ideas of a vector field, without any coordinates involved.

   Finally, we can define “arrows” at each “point” of the manifold, as in our original picture of the sphere and arrows tangent to points on it.  We will just evaluate the function returned by the vector field at the relevant point , i.e. .  Defining the tangent vector by , we think of the real number as the result of differentiating the function in the direction of the tangent vector .  We define the tangent space at , , to be the set of all .

   We can see that our original picture of identifying vectors with their components stems from the nature of the tangent space of .  Since for any point , the tangent vectors form a basis, we sloppily identify two vectors and at different points to be the same if .  However, in actuality and are vectors in different tangent spaces, even if .  We can get away with this sloppiness in , but not when dealing with more complex manifolds.

   In any case, this coordinate-free definition of vector fields allows us to define even more structures on the manifold.  These will be useful in distilling the essential qualities of Maxwell’s Equations.

SO(3), SU(2), and the Dirac Equation

   Today I read a bit of Ryder’s QFT (which was $20 when I bought it on Amazon; a day after I bought it, it dropped to $14 (!), but it is now again $53?), starting over with Chapter 2.  Ryder approaches relativistic wave equations from symmetry arguments, which I like.  Below are some notes to myself on his introduction of the Dirac equation.

   First, he demonstrates the problems with the Klein-Gordon equation for scalar particles,

    

   The first problem is obvious; this equation is obtained by the substitution of differential operators for and , as usual in quantum theory, into the relativistic energy-momentum relation .  But this relation gives both positive and negative energy solutions, .  These negative energy solutions are the first problem.

   The second problem stems from attempting to modify the probabilistic interpretation of the wave function into one consistent with relativity.  Taking the usual non-relativistic probability density, , and attemping to make it the time component of a 4-vector

     ,

where

      

is the probability current, gives

    

   However, since the Klein-Gordon equation is second-order in time, initial conditions for and can be chosen independently, and thus is not guaranteed to be positive-definite – which makes a probabilistic interpretation difficult!

   To fix these problems, Dirac introduced his first-order equation.  Now, some texts, like Griffiths’s Elementary Particles book, attempt to introduce the Dirac equation by “factoring” the Klein-Gordon equation, and then showing that these factors must necessarily imply a first-order matrix equation – i.e., the Dirac equation.  Ryder does this too, but first derives the Dirac equation by considering the transformation of spinors under groups isomorphic to the Lorentz group.  In this way, the Dirac equation is given by symmetry principles.

   Even before showing this, as an analogy Ryder demonstrates that SU(2) is a double cover of SO(3).  I also found a good discussion of this here (there is also more detailed discussion of group representations and the Lorentz group in Maggiore and in Aitchison’s notes).  Basically, one can show that any element of SO(3), the group of orientation-preserving rotations of a 3-dimensional vector space, can be mapped to two elements of SU(2), the group of unitary transformations of a 2-dimensional complex spinor space.  This two-to-one mapping derives from the similar-but-not-identical Lie algebras of the two Lie groups, given by the commutator relations obeyed by the generators of the groups: for the angular momentum operators that generate SO(3) vs. for the Pauli matrices that generate SU(2).

   Likewise, the generators of the Lorentz group are similar to those of other groups; however, in this case, the similarity must be made explicit.  To see this, first notice that the set of pure Lorentz transformations, the generators of which we shall call , is not closed; e.g., the group commutator of boosts in the x and y directions gives a rotation in the z direction.  Thus, the true Lorentz group includes both boosts and rotations.  That is, the Lie algebra relations of the Lorentz group intertwine both and , like so:

     ,
     ,

along with the usual commutator for rotations.

   Now, we can write complex linear combinations of these 6 generators to form 6 new generators for the complexified Lorentz group:

     . 

   Then, separately satisfy the commutation relation of the SU(2) group, so the Lorentz group SO(3,1) is roughly analogous to SU(2)SU(2).  The isomorphism is not exact because of the complexification; however, we can use this similarity to study the transformation effects of boosts on the two different types of spinors associated with each SU(2) group (defining the spinor space on which SU(2)SU(2) is represented as ).  It can also be shown that these two spin- spaces switch under parity (basically because is an axial vector, while is a true vector).  If we wish to include parity in our equations, it becomes necessary to consider both types of 2-component spinors, which transform differently under boosts, combined as one 4-component spinor.

   By studying how this 4-component Dirac spinor transforms under boosts, one finds the Dirac equation.  Essentially, one boosts a rest frame Dirac spinor, and then uses the equivalence of both rest frame 2-component spinors to write relations between boosted 2-component spinors of both types.  This relation can be cast back into an equation involving a set of 4×4 matrices   and the Dirac spinor , which is the Dirac equation:

     . 

   The obey the anti-commutation relation .  Of course, one could have found this relation by “factoring” the Klein-Gordon equation,

     ,

finding that the and are the matrices, but this is far less revealing of the rich structure of the Dirac spinor representation of the Lorentz group.

   Now the Dirac equation is first-order, and a positive-definite probability density can be found.  Negative energy solutions still exist, but because of the fermionic nature of the spin- particles described by the equation the Pauli exclusion principle can be invoked to prevent a particle from falling to infinitely negative energy.  However, for the exclusion principle to fix this problem requires that the negative energy states be filled by an infinity of particles – the so-called Dirac sea.  Dirac postulated that antiparticles would appear as “holes” in this sea, predicting the existence of the positron.  Despite the success of this prediction, this somewhat awkward picture of the Dirac sea is voided by the full QFT formulation (which allows bosonic antiparticles, despite the lack of a bosonic exclusion principle, as well); however, even QFT requires its own sea of infinite energy – i.e., the zero-point energy of the vacuum.

   Thus, by extending the Lorentz group by parity and finding how the Dirac spinors on which the group is represented transform under boosts, one can derive the Dirac equation from group theoretic and symmetry arguments!

   Kind of sketchy for the first post of real substance, but this was mostly to check out the capabilities.  I wrote this post bit-by-bit over a few days, and have since then branched off from Ryder to peruse Maggiore and Aitchison and Hey.  I actually think I will make Aitchison and Hey my text of choice; the presentation is clearer than Ryder, if less interesting, and there are problems given in the book and solutions online at Aitchison’s website (see above for relevant links).  Nevertheless, I hope to find a clearer, mathematically rigorous presentation of the isomorphisms and relations between the various Lie groups in the future.  This looks like a good overview.

First Post!

   Hello, this is my physics blog.  I decided to start a physics blog so I could blog about physics.  Tautological!  I chose WordPress for my blog so I could use to write equations.  I named my blog “on the seashore” for two reasons: 1) because of the above quote by Sir Isaac Newton, and 2) because I have just completed my physics undergraduate education on one coast, and am about to begin my physics graduate education on another coast.

   For this summer, I will be reviewing some old physics and learning some new physics, to prepare for entrance exams and doing research.  Mostly I will blog notes to myself to sort ideas out and to make sure I’m understanding things.  Here’s a list of topics and books I hope to get through:

Classical Mechanics: I think I will browse through Landau and Lifshitz again to refresh my memory.
Electrodynamics: Griffiths should be fine.
Quantum Mechanics: Sakurai should do the trick here. I should probably learn some of the approximations and scattering that I skipped over during my undergrad.
Statistical Mechanics: The stat mech course at my undergrad school was awful. So I am going to read Bloch’s notes to patch up the damage.
Quantum Field Theory: I have had difficulty finding a QFT text that I like. I started with Zee, which I found interesting but far too unstructured. Next I tried Weinberg, which was too heavy on unconventional math and Weinberg’s own formalisms. I then took a course that used Peskin & Schroeder, which I find light on physical motivation and heavy on phenomenological calculations; it is also lacking in important examples - the complex scalar field is left as an exercise to the reader, and took me quite a few pages to work through!  I like Lahiri & Pal, but unfortunately don’t have my own copy. I do have a copy of Ryder, which will be my text of choice for the summer. I will supplement it with Mark Srednicki’s notes, Griffiths’s Elementary Particles book, Aitchison & Hey, and Maggiore.
General Relativity: I’ll review Carroll’s book and try to tackle Wald.
Cosmology: I’ll review notes from a course on the Cosmic Microwave Background that I took last fall semester. I hope to do research in this area, so I’d better brush up!
Mathematical Physics, etc.: I’m going to make up for my spotty theoretical mathematics background by going through Geroch’s book. I also recently was given a copy of Baez & Muniain, which I like very much and will try to complete! I also got a third of the way through Zwiebach’s string theory book during this past semester, so I will try to finish that up.
Problems: I have copies of Cahn & Nadgorny’s two compilations of physics problems taken from graduate school qualifiers. I should work through some problems, since I haven’t really touched pen and paper in a whole semester!