Superspace

It has been a while since I posted anything on this WordPress blog. Since I’ve just bought a domain name from WordPress, I thought I should post something, even if just to mention this update in the blog’s address. But that’s just some filler content that I post on my other blog. So here I should at least talk about some Physics.

Superspace – the greatest invention since the wheel – is where I spent most of my graduate days. After graduating, I have spent less time there but I have regularly visited it over the years. Last year, I visited it to settle a 7 (or so) years old score with my collaborators by writing a paper on quantization of supersymmetric Yang-Mills theory in 4d \mathcal{N}=3 harmonic superspace that is actually useful in computing loop graphs.

Now, a popular physics blog would go to great lengths to try to explain to a layperson the various technical words that appear in the above paragraph, like quantization, SYM theory, \mathcal{N}=3, harmonic superspace, and loop graphs. [Of course, 4d just refers to our (3+1)-dimensional spacetime: 3 space and 1 time directions.] As it turns out, 4Gravitons is just that sort of a blog and most of these terms have been explained there: Quantization, Supersymmetric, Yang-Mills theory, \mathcal{N}=3 (well, \mathcal{N}=3 is quite similar to \mathcal{N}=4), Loop graphs.

The most astute reader amongst you must have already realized that harmonic superspace is not explained there. Or maybe, it has been but I have deliberately not linked to that page. The reason for doing that is quite simple: I am supposed to be the (second-)last ‘expert’ on harmonic and projective superspaces in the whole world. By “(next-to-)last ‘expert’ in the whole world”, I don’t mean the pejorative phrase “world-famous in India”, I literally mean the former phrase and it does not have any connotation about fame at all in it. So, being one of the ‘experts’ in superspace, I can’t link everything to 4Gravitons (who is an expert in another subfield of theoretical high energy physics) as I believe that one should undertake the undertaking about explaining something that one is an expert on to the lay-world.

Here goes my feeble attempt then. [Disclaimer: Anyone expecting clarity and thoughtfulness in crafting the explanations, as found at 4Gravitons, will be disappointed.] First of all, we must agree with the basic fact that harmonic superspace is a kind of a superspace. Then, we need to define a superspace and later we will tag on the adjective ‘harmonic’ in a hand-waving way, which should be sufficient for the level of this post.

A superspace is an extension of the usual spacetime x^{\mu}=(t,x,y,z) with extra coordinates which have no basis in reality. (There, I said it; no beating around the bush.) These coordinates are such that (let’s call them θ (theta) to exude Greek sophistication) \theta^2=0. Now, a layperson would say: well, that just means \theta=0. That is, a layperson who has never heard of matrices. Anyway, hoping for this level of mathematical sophistication from our lay-audience, we can get past the fact that the vanishing square is not a problem and we can get on with building the most basic superspace, called \mathcal{N}=1 superspace. This superspace has 4 θ’s [\theta_1\theta_2=-\theta_2\theta_1 is satisfied by multiple θ’s. Recall that ordinary numbers satisfy x_1x_2=+x_2x_1] and can be used to describe \mathcal{N}=1 (and higher) supersymmetric theories. This is useful because with this setup, supersymmetric theories can be described in a compact notation with very concise expressions. Performing calculations in superspace is also quick and easy when compared to ordinary spacetime computations. Before superspace was invented, people found “miraculous cancellations” in their calculations, i.e., they would calculate a bunch of Feynman graphs and end up with an equivalent of “+1-1=0”. The problem is that they would have to compute the equivalents of “+1” and “-1” separately (which can be tedious) and find later that they summed to 0. Whereas in superspace, there are far fewer (super)Feynman graphs to compute and one directly sees the 0’s. In other words, nothing tedious to calculate and cancel later.

Now, we come to another fact that there are \mathcal{N}=2 supersymmetric theories and we should probably have \mathcal{N}=2 superspace to describe them directly. So one may just append 4 more θ’s to \mathcal{N}=1 superspace and call it a day. That approach is a valid one but doesn’t go far in achieving all the successes of \mathcal{N}=1 superspace in simplifying computations for \mathcal{N}=2 theories. (Because there are certain objects that do not exist in this naïve \mathcal{N}=2 superspace, which turn out to be crucial for simplifying computations.) Thus, we have to invoke harmonic (or projective) variety of the \mathcal{N}=2 superspace. Harmonic superspace contains the usual spacetime, only 4 θ’s (which are obtained as some combinations of the earlier 8 θ’s) and a 2d sphere (an ordinary sphere, nothing exotic like the θ’s but it is again not ‘real’; though, it has a very satisfying origin story in the mathematical structure of supersymmetric theories, whose discussion is beyond the scope of this post). It might still not be clear to a layperson what has ‘harmonic’ got to do with that extra 2d sphere. The punchline being that the study of functions on spheres is referred to as harmonic analysis in the mathematical literature. (The adjective ‘projective’ is also related to the 2d sphere, which can be treated as a complex projective line. Actually, the word projective has a more satisfying origin in the way the coordinates transform in projective superspace. You guessed it right: they transform via projective transformations.) QED

Just completing the loop, \mathcal{N}=3 harmonic superspace consists of the usual spacetime, 8 θ’s (instead of the expected 12) and a 6d compact space (don’t really want to say anything more than that right now). Our 7-years-in-making paper studies the SYM theory in this superspace and derives Feynman rules to make computations of loop graphs easier. This problem has been unsolved for more than 3 decades. One proof-of-concept example we include in our paper slashes a 6-pages computation based on symmetry arguments to a ½-page direct computation of two very similar Feynman graphs.

1 Loop 4 Point Graphs

So I guess that’s all for this post. Till next time, subscribe to 4Gravitons. You will learn a lot about Theoretical High Energy Physic(icist)s.

Featured

hephys Bibliography Style

[Update(08/08/19): Changes noted below are reflected in the original text further down.

  1. DOI hyperlinking for inCollection and inProceedings/Conference has been changed from “chapter, pages” to “title”.
  2. Made “Chapter” optional for inCollection.

End of ChangeLog.]

You should have come from here.

Reiterating, I have created ‘hephys’ bibliography style which can be used to add references in hep-th papers. I was planning to compare hephys with “utphys” (my preferred style before I thought of creating my own) but that may leave a bad taste in some mouths including my own so let me just highlight some features of hephys, with no ulterior motive. hephys is optimized for InSpire’s BibTeX data so it’s mostly intended for people writing hep-th papers. In fact, utphys is quite general so anyone not writing hep-th papers should just stick to it.

Some interesting facts about hephys:

  1. hephys tries to follow mostly LaTeX(EU) style typesetting of citations.
  2. hephys correctly typesets and links both old-style and new-style arXiv-identifiers.
  3. hephys treats url field as doi if doi field is not present.
  4. If doi field is present, then hephys appends url at the end of the citation only if it is a different link (it can’t be an arXiv link either; as I said before, very specialized to hep-th papers).
  5. hephys clubs number and volume together as in vol [num] for journal articles.
  6. After everything’s thought and written, there’s a possible deal-breaker: “Title”, vs. “Title,”. (No prizes for guessing which one hephys outputs!😆)

On to some history now: I used ‘makebst’ command (available in any LaTeX installation) to create a skeleton style file ages ago. Why was it a skeleton, you ask? Because it had no (when I say no, I mean NO) support for hyperlinking or any idea for handling arXiv data properly. So where do we go from there? I don’t know about ‘we’ but I learnt that .bst files are written in an unnamed programming language using postfix notation! That brought back ‘great’ memories from class IX (nearly 2 decades ago) when we were taught some ‘computer theory’ along with BASIC (a basic programming language). Never thought that there would be a ‘serious’ language based on this notation. I have seen some discussion about prefix / infix / postfix notation in Mathematica somewhere but still as a curiosity rather than in any actual programming usage. But, here there was an opportunity to explore the thing that I once thought was ‘unexplore-able’. Considering myself a programming expert, I was able to edit a few things to make arXiv links appear as I wanted, after going through this manual half a dozen times.

Then I remembered there’s something called DOI which can be used to hyperlink journal info! That was the time to open utphys.bst file and steal Jacques Distler’s code (basically the function “add.doi”). My expert senses, not to rest so easily, went on to give me sleepless nights till I fixed the behaviour of url field as highlighted in points 3 & 4 above. Oh boy! writing if-else and for-loop in postfix notation is not child’s play (read: mind-numbingly frustrating) but wow, I’d be lying if I didn’t say It’s Quite Fulfilling! Also, I made very few keywords mandatory for a given citation style so not many errors pop up even if the BibTeX data is not up to the mark (i.e., cobbled up while in a half-awake state at midnight without any regards for health & safety instructions). After that, to reduce the chances of hephys clashing with utphys too much, I removed a few things to make it unappealing to people writing non-hep-th papers. The following are the only citation styles (with the accepted / expected keywords; only bold ones are required) supported in hephys and their sample outputs (the blue colouring just means the expected hyperlink as explained in points 2, 3 & 4 above):

1. Article: author, collaboration, title, journal, volume, number, year, pages, doi, note, eprint, primaryClass, url.

Samples:

@article{art:1, author = “author”, collaboration = “collaboration”, title = “article with both doi and url”, journal = “journal”, volume = “vol”, number = “num”, year = “year”, pages = “pages”, doi = “doi”, note = “note”, eprint = “eprint”, primaryClass = “pClass”, url = “⋯.url.⋯”}

[1] author (collaboration), “article with both doi and url”, journal vol[num] (year) pages, note, arXiv:eprint [pClass], URL.

@article{art:2, author = “author”, collaboration = “collaboration”, title = “article with both doi and url but duplicate url not used”, journal = “journal”, volume = “vol”, number = “num”, year = “year”, pages = “pages”, doi = “doi”, note = “note”, eprint = “pClass/eprint”, primaryClass = “pClass”, url = “····://··.doi.···/10.1086⋯x”}

[2] author (collaboration), “article with both doi and url but duplicate url not used”, journal vol[num] (year) pages, note, arXiv:pClass/eprint.

@article{art:3, author = “author”, collaboration = “collaboration”, title = “article with only url used as doi”, journal = “journal”, volume = “vol”, number = “num”, year = “year”, pages = “pages”, note = “note”, eprint = “eprint”, primaryClass = “pClass”, url = “⋯.url.⋯”}

[3] author (collaboration), “article with only url used as doi”, journal vol[num] (year) pages, note, arXiv:eprint [pClass].

@article{art:4, author = “author”, collaboration = “collaboration”, title = “article with neither doi nor url”, journal = “journal”, volume = “vol”, number = “num”, year = “year”, pages = “pages”, note = “note”, eprint = “pClass/eprint”, primaryClass = “pClass”}

[4] author (collaboration), “article with neither doi nor url”, journal vol[num] (year) pages, note, arXiv:pClass/eprint.

@article{art:5, author = “author”, collaboration = “collaboration”, title = “article with no journal”, year = “year”, pages = “pages”, note = “note”, eprint = “pClass/eprint”, primaryClass = “pClass”}

[5] author (collaboration), “article with no journal”, year, note, arXiv:pClass/eprint.

2. Book: author/editor, collaboration, title, doi, volume/number, series, edition, publisher, address, year, note, eprint, primaryClass, url.

Samples:

@book{boo:1, author = “author”, collaboration = “collaboration”, title = “book”, doi = “doi”, number = “num”, series = “series”, edition = “edn”, publisher = “publisher”, address = “address”, year = “year”, note = “note”, eprint = “eprint”, primaryClass = “pClass”, url = “⋯.url.⋯”}

[6] author (collaboration), book, number num in series, edn edition, publisher, address year, note, arXiv:eprint [pClass], URL.

@book{boo:2, editor = “editor”, title = “book”, doi = “doi”, volume = “vol”, series = “series”, edition = “edn”, publisher = “publisher”, address = “address”, year = “year”, note = “note”, eprint = “eprint”, primaryClass = “pClass”, url = “⋯.url.⋯”}

[7] editor (Ed.) book, volume vol of series, edn edition, publisher, address year, note, arXiv:eprint [pClass], URL.

3. Booklet: author, collaboration, title, howpublished, doi, year, note, eprint, primaryClass, url.

Sample:

@booklet{boo:3, author = “author”, collaboration = “collaboration”, title = “booklet”, howpublished = “howpublished”, doi = “doi”, address = “address”, year = “year”, note = “note”, eprint = “eprint”, primaryClass = “pClass”, url = “⋯.url.⋯”}

[8] author (collaboration), “booklet”, howpublished, address year, note, arXiv:eprint [pClass], URL.

4. inBook: author/editor, collaboration, title, volume/number, series, chapter, pages, doi, edition, publisher, address, year, note, eprint, primaryClass, url.

Sample:

@inbook{inb:1, editor = “editor”, title = “inbook”, volume = “vol”, series = “series”, chapter = “ch”, pages = “pages”, doi = “doi”, edition = “edn”, publisher = “publisher”, address = “address”, year = “year”, note = “note”, eprint = “eprint”, primaryClass = “pClass”, url = “⋯.url.⋯”}

[9] editor (Ed.), inbook, volume vol of series, chapter ch, p. pages, edn edition, publisher, address year, note, arXiv:eprint [pClass], URL.

5. inCollection: author, collaboration, title, booktitle, editor, edition, chapter, pages, doi, volume/number, series, publisher, address, year, note, eprint, primaryClass, url.

Sample:

@incollection{inc:1, author = “author”, collaboration = “collaboration”, title = “incollection”, booktitle = “booktitle”, editor = “editor”, edition = “edn”, chapter = “ch”, pages = “pages”, doi = “doi”, number = “num”, series = “series”, publisher = “publisher”, address = “address”, year = “year”, note = “note”, eprint = “eprint”, primaryClass = “pClass”, url = “⋯.url.⋯”}

[10] author (collaboration), “incollection“, in booktitle (edited by editor), edn edition, chapter ch, p. pages, number num in series, publisher, address year, note, arXiv:eprint [pClass], URL.

6. inProceedings/Conference: author, collaboration, title, booktitle, volume/number, series, editor, pages, doi, organization, publisher, address, year, note, eprint, primaryClass, url.

Sample:

@inproceedings{inp:1, author = “author”, collaboration = “collaboration”, title = “inproceedings”, booktitle = “booktitle”, volume = “vol”, series = “series”, editor = “editor”, pages = “pages”, doi = “doi”, organization = “organization”, publisher = “publisher”, address = “address”, year = “year”, note = “note”, eprint = “eprint”, primaryClass = “pClass”, url = “⋯.url.⋯”}

[11] author (collaboration), “inproceedings“, in booktitle, volume vol of series (edited by editor), p. pages, organization, publisher, address year, note, arXiv:eprint [pClass], URL.

7. MastersThesis: author, collaboration, title, doi, school, address, year, note, eprint, primaryClass, url.

Sample:

@mastersthesis{mas:1, author = “author”, collaboration = “collaboration”, title = “mastersthesis”, doi = “doi”, school = “school”, address = “address”, year = “year”, note = “note”, eprint = “eprint”, primaryClass = “pClass”, url = “⋯.url.⋯”}

[12] author (collaboration), mastersthesis, Master’s thesis, school, address year, note, arXiv:eprint [pClass], URL.

8. Misc: author, collaboration, title, howpublished, url, year, note.

Sample:

@misc{mis:1, author = “author”, collaboration = “collaboration”, title = “misc”, howpublished = “howpublished”, url = “⋯.url.⋯”, year = “year”, note = “note”}

[13] author (collaboration), “misc”, howpublished year, note.

9. PhdThesis: author, collaboration, title, doi, school, address, year, note, eprint, primaryClass, url.

Sample:

@phdthesis{phd:1, author = “author”, collaboration = “collaboration”, title = “phdthesis”, doi = “doi”, school = “school”, address = “address”, year = “year”, note = “note”, eprint = “eprint”, primaryClass = “pClass”, url = “⋯.url.⋯”}

[14] author (collaboration), phdthesis, Ph.D. thesis, school, address year, note, arXiv:eprint [pClass], URL.

10. Proceedings: editor, title, doi, volume/number, series, organization, publisher, address, year, note, eprint, primaryClass, url.

Sample:

@proceedings{pro:1, editor = “editor”, title = “proceedings”, doi = “doi”, volume = “vol”, series = “series”, organization = “organization”, publisher = “publisher”, address = “address”, year = “year”, note = “note”, eprint = “eprint”, primaryClass = “pClass”, url = “⋯.url.⋯”}

[15] editor (Ed.), proceedings, volume vol of series, organization, publisher, address year, note, arXiv:eprint [pClass], URL.

11. Unpublished: author, collaboration, title, year, note.

Sample:

@unpublished{unp:1, author = “author”, collaboration = “collaboration”, title = “unpublished”, year = “year”, note = “note”}

[16] author (collaboration), “unpublished”, year, note.

So there you have it, the full ‘manual’ for hephys. If you find any bugs, do send me the offending citation’s BibTeX data & expected output and I’ll look into it. There is one bug regarding citations having erratum/addendum information and two doi’s. It’s not handled correctly for now but I’m thinking of doing something about it in the new year. On second thought, there may not be a universal format for such data so I’m a bit hesitant to start looking into separating multiple doi’s and journal ref’s just yet. Anyway, if you (yes You, the zeroth reader of this post) are up for programming these and more fixes in your own ‘hephys’, do let me know. After all, the world can only rejoice with more choices to consider!‍🤦‍♂️

Download Link for those who reached this point!

Projective Transformations

Today’s problem is how to transform a given quadrilateral to another quadrilateral.

Let me be more precise. The transformation is to be considered in the context of photography. For example, look at this ‘Scrabble’ board.

Quadrilateral

As one might have expected for such a board to look like a square (instead of that unsightly trapezoid), it’s our job in this post to ‘correct’ the above image. That is, to transform a given quadrilateral to a ‘proper’ rectangle.

The solution comes in the form of Projective transformation in 2D. Projective transformations are basically fractional linear transformations:

x^{\prime}=\frac{ax+by+c}{gx+hy+i},\quad y^{\prime}=\frac{dx+ey+f}{gx+hy+i}.

Why isn’t just a simple linear transformation enough for this job? Because as we know, a 2D matrix can only describe scaling, rotation and shear. It doesn’t have translations and it also can NOT make converging lines parallel! So, not only linear transformation is insufficient, even affine transformation won’t fit our purposes. The transformation which can make parallel lines converge (i.e. make ∞ come to a finite point) is what we are really after and the projective one does that job. Still all is not lost. All these 5 transformations can be made linear if we are willing to go up a bit from 2D to 3D.

Let’s augment our 2D coordinates (x,y) by w and consider the following 3D linear transformation:

\begin{pmatrix}x^{\prime} \\ y^{\prime} \\ w^{\prime}\end{pmatrix}=\begin{pmatrix}a & b & c \\ d & e & f \\ g & h & i\end{pmatrix}\begin{pmatrix}x \\ y \\ w\end{pmatrix}

To get the 2D coordinates, we have to project back from 3D by dividing (x, y) by w. Since that division is a bit cumbersome, we will choose w=1. (The division makes sure w^{\prime}=1 too.) Let’s do a sanity check: a, b, d, e give the usual linear (2D) transformations, c, f give the translations, g, h give the projective transformations, and i is a global scaling. This last transformation is redundant for our purposes so we can set i=1.

Now, we have the right transformation tool. Why, you may ask? The answer is because this transformation matrix has 8 (unknown) parameters and given the 4 corners of the (source) quadrilateral & (destination) rectangle, we can write down 8 equations relating them. So 8 equations and 8 unknowns → any respectable linear algebra package should be able to do the ‘Maths’! (Caveat: You – How do you know the destination corners? Me – Well, that’s ‘beyond the scope’ of this post. You – $$$##%%^#)

Actually, we can do better than that. If one stares at those 8 equations long enough, one realizes that those equations can be solved analytically if the transformation is from a unit square to a given quadrilateral! You might be thinking that’s not too helpful; we want the reverse at the very least. Not quite! Because in image transformation business, if you think about it, “you don’t PUT the pixel, you GET the pixel”. (That’s my quote and you can fearlessly attribute it to me from now on. – Thanks.) For more discussion on this revelation, read this post.

So what we have to do now is simple:

  1. Get the 4 coordinates of the corners of the quadrilateral in the source image: X_q .
  2. Find the 8 parameters (a, \cdots, h) in the transformation matrix from the analytical solution for the coordinates X_q: T_s .
  3. Figure out the corners of the destination rectangle: X_r .
  4. Scale the transformation matrix in (x,y) direction appropriately so that the square scales to the required rectangle: T_r=T_s \cdot\text{diag}\left(\tfrac{1}{W}, \tfrac{1}{H}, 1\right) . (Width & Height are figured out from X_r.)
  5. Do the final transformation with translated coordinates: X_{\text{src}}=T_r \cdot (X_{\text{dest}}-X_{\text{trans}}) . (I find that incorporating the translations in the matrix is not that straightforward. Maybe it can be done, but translating the coordinates and then transforming them is simple enough. Also remember: “GET the Pixels”.)
  6. Crop the relevant portion of the transformed image.

After implementing the above algorithm, we can end up with something like this (I think the second last point becomes clear too):

Intermediary

Or to put it more bluntly, this:

Square

If you’re starting to think, I thought of all this… You’re giving me too much credit. Here’s the paper from where I learnt about this solution / algorithm (though, I think the solution given there may have some typos. I say this because that solution didn’t work ‘out of the box’):

Source

A Difficult Problem?

Many people land on this page of my other blog in search of an answer for this problem:

A circular lake 1.0 km in diameter is 10 m deep. Solar energy is incident on the lake at an average rate of 200 W/m². If the lake absorbs all this energy and does not exchange heat with its surroundings, how long will it take to warm from 10°C  to 20°C ?

as seen in the following snapshot:

Problem

Now, I do offer a pathway to the solution there but I feel that not providing the final answer is tantamount to cheating on my part. So, this being a ‘Physics blog’, I provide the entire solution here (I don’t know or care what good does it do to those ‘searchers’!):

t=\frac{Q}{S.A}=\frac{m.C.\Delta T}{S.A}=\frac{(\rho .V)C.\Delta T}{S.A}=\frac{\rho(A.h)C.\Delta T}{S.A}=\frac{\rho .h.C.\Delta T}{S}

\Rightarrow t=\frac{1000\times 10\times 4200\times (20-10)}{200}= 2.1\times 10^6s=24d~7h~20m.

3D Rotations

You should have come from here!

Here’s a quickie: What are the eigenvalues of a 2D rotation matrix?

Here’s a problem: For a bunch of rotations performed one after another on a 3D object, find an equivalent single rotation which would give the same result.

Here’s a solution: First of all, multiply all the rotation matrices (obviously!) such that R_{s}=R_{n}R_{n-1}\cdots R_{2}R_{1}. A single rotation is characterized by an axis and an angle of rotation. Let’s get the angle first. Find the eigenvalues of R_{s}. Two of them will be of the form: e^{\dot{\iota}\theta}\,\&\,e^{-\dot{\iota}\theta}. This θ is the required angle. Let’s get the axis now. So what would be the third eigenvalue? Remember R_{s} \in SO(3) which means the third one is 1! Find the eigenvector corresponding to this eigenvalue which is the required axis!

Here’s an example: Let’s do the problem mentioned in Gravitation.

R_{1}=\begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix};

R_{2}=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \end{pmatrix}.

\Rightarrow R_{s}=R_{2}R_{1}=\begin{pmatrix} 0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0 \end{pmatrix}.

Relevant eigenvalues of R_{s} are -\frac{1}{2}\pm\frac{\sqrt{3}}{2}\dot{\iota} which give us the angle: \mathrm{tan}^{-1}\left(-\sqrt{3}\right)=120^{\circ}. The eigenvector corresponding to 1 is \frac{1}{\sqrt{3}}(1,-1,1) so the axis is one of the diagonals! This solution agrees with the one given in the book.

Physical DoF of Graviton & Photon(revised)

1) DoF of Graviton or more precisely, modes of g_{\mu\nu}(x) (because nothing is quantized here!):

Let us start by writing g_{\mu\nu}(x)=\eta_{\mu\nu}+\kappa h_{\mu\nu}(x), where \kappa^2=\frac{8\pi G}{c^4}. With this definition, Einstein’s field equation at linearized level becomes:

G_{\mu\nu}^{lin}=-\kappa^2 T_{\mu\nu}^{(2)}

where the RHS refers to quadratic part of the Energy-Momentum tensor. For counting physical modes, we need to consider gravitational field without any sources so we will deal with just[1,2]:

G_{\mu\nu}^{lin}=\frac{\kappa}{2}\left[\square h_{\mu\nu}-h_{\mu,\nu}-h_{\nu,\mu}+h_{,\mu\nu}-\eta_{\mu\nu}\left(\square h-h_{\alpha}^{,\alpha}\right)\right]=0

where h_{\mu}=\partial^{\nu}h_{\mu\nu}~\mathrm{and}~h=\eta^{\mu\nu}h_{\mu\nu}. This linearized field equation is invariant under the following gauge transformation:

\delta h_{\mu\nu}=\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}

which is just the linearized version of the Einstein (general coordinate) transformation of the metric, \delta g_{\mu\nu}=\xi^{\alpha}\partial_{\alpha}g_{\mu\nu}+g_{\alpha\nu}\partial_{\mu}\xi^{\alpha}+g_{\mu\alpha}\partial_{\nu}\xi^{\alpha}!

Now, as is usually done, we choose a gauge called de Donder gauge[3]: \partial^{\mu}\left(h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h\right)=0. With this gauge choice and defining \phi_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h, we get the following redefined equations:

Field Equation: G_{\mu\nu}^{lin}=\frac{\kappa}{2}\square \phi_{\mu\nu}=0

de Donder Gauge: \partial^{\mu}\phi_{\mu\nu}=0

Gauge Transformation: \delta\phi_{\mu\nu}=\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}-\eta_{\mu\nu}\partial_{\alpha}\xi^{\alpha}

Let us choose a plane wave solution for the field equation: \phi_{\mu\nu}=\epsilon_{\mu\nu} e^{i k.x}. First thing to note is that \phi_{\mu\nu} has 10 components/modes (in 4-D obviously!). The question is how many of them are physical modes. You probably already know the answer but we’ll get there slowly but concretely! Let us plug the solution in the first two equations above to get:

\square\phi_{\mu\nu} = -k^2\phi_{\mu\nu}=0 \Rightarrow k^2=0

\partial^{\mu}\phi_{\mu\nu}=0 \Rightarrow k^{\mu}\epsilon_{\mu\nu}=0

The last equation reduces the number of independent modes in \phi_{\mu\nu} to 10-4=6. We have the following 6 ‘most general’ basis tensors for \epsilon_{\mu\nu} assuming wave propagation is in 3(z)-direction, i.e. k_{\mu}=(-1,0,0,1):

1.~\epsilon_{1\mu}\epsilon_{1\nu}+\epsilon_{2\mu}\epsilon_{2\nu},

2.~\epsilon_{1\mu}\epsilon_{1\nu}-\epsilon_{2\mu}\epsilon_{2\nu},

3.~\epsilon_{1\mu}\epsilon_{2\nu}+\epsilon_{1\nu}\epsilon_{2\mu},

4.~\epsilon_{1\mu}k_{\nu}+\epsilon_{1\nu}k_{\mu},

5.~\epsilon_{2\mu}k_{\nu}+\epsilon_{2\nu}k_{\mu} and

6.~k_{\mu}k_{\nu}.

where \epsilon_{1\nu}=\delta^{\mu}_{1}\epsilon_{\mu\nu} and so on. Let us now look at the gauge transformation of \epsilon_{\mu\nu} after choosing a specific form of the gauge parameter, \xi_{\mu}=C a_{\mu}e^{i k.x} (where C is a constant which could be absorbed in a_{\mu}!):

\epsilon_{\mu\nu}^{\prime} = \epsilon_{\mu\nu}+i C\left(a_{\mu}k_{\nu}+a_{\nu}k_{\mu}-\eta_{\mu\nu}a_{\alpha}k^{\alpha}\right).

As we did for \epsilon_{\mu\nu}, the most general basis vectors for a_{\mu} are chosen to be \epsilon_{1\mu}, \epsilon_{2\mu}, k_{\mu}~\&~\bar{k}_{\mu}=(1,0,0,1). We immediately see that the choice of first three basis vectors and a ‘correct’ C makes the modes 4, 5 & 6 pure gauge i.e. \epsilon^{\prime}=0. Last basis vector is a bit tricky, so let us look at this tricky identity:

\eta_{\mu\nu}=\epsilon_{1\mu}\epsilon_{1\nu}+\epsilon_{2\mu}\epsilon_{2\nu}+\frac{\bar{k}_{\mu}k_{\nu}+\bar{k}_{\nu}k_{\mu}}{\bar{k}_{\alpha}k^{\alpha}},

And you immediately ‘see’ that (with ‘correct’ C) the mode which is gauged away is 1! So we are left with 6-4=2 physical modes:

\epsilon^{+}_{\mu\nu}=\epsilon_{1\mu}\epsilon_{1\nu}-\epsilon_{2\mu}\epsilon_{2\nu}~\mathrm{and}~\epsilon^{\times}_{\mu\nu}=\epsilon_{1\mu}\epsilon_{2\nu}+\epsilon_{1\nu}\epsilon_{2\mu}.

These are the two ‘famous’ plus- & cross- modes/polarizations of the gravitational waves.

***

2) DoF of Photon or more precisely, modes of A_{\mu}(x):

The Maxwell’s equation of motion is as follows:

\square A_{\mu}-\partial_{\mu}\partial^{\nu}A_{\nu}=0.

This equation is invariant under the following gauge transformation:

\delta A_{\mu}=\partial_{\mu}\lambda.

Now, we choose a gauge called Lorentz gauge:

\partial^{\mu}A_{\mu}=0.

With this gauge choice, we get the following simplified equation of motion:

\square A_{\mu}=0.

Let us choose a plane wave solution for this equation: A_{\mu}=\epsilon_{\mu} e^{i k.x}. Firstly, A_{\mu} has 4 components/modes (in 4-D!). Plugging this solution in the two equations above, we get:

\square A_{\mu} = -k^2 A_{\mu}=0 \Rightarrow k^2=0

\partial^{\mu}A_{\mu}=0 \Rightarrow k^{\mu}\epsilon_{\mu}=0

The last equation reduces the number of independent modes in A_{\mu} to 4-1=3. We have the following 3 ‘most general’ basis vectors for \epsilon_{\mu} (assuming wave propagates in 3(z)-direction):

1.~\delta^{1}_{\mu},

2.~\delta^{2}_{\mu} and

3.~k_{\mu}.

Let us now look at the gauge transformation of \epsilon_{\mu} after choosing a specific form of the gauge parameter: \lambda=C e^{i k.x} (where C is a constant!) which gives:

\epsilon_{\mu}^{\prime} = \epsilon_{\mu}+i C k_{\mu}.

We immediately see that a ‘correct’ C makes the mode 3 pure gauge. So we are left with 3-1=2 physical modes:

\epsilon^{x}_{\mu}=\delta^{1}_{\mu}~\mathrm{and}~\epsilon^{y}_{\mu}=\delta^{2}_{\mu}

These are the two ‘famous’ linear modes/polarizations of the electromagnetic waves.

———

[1] If you are wondering which action gives that field equation, it is just the quadratic part of the Hilbert-Einstein action (which is proportional to the Fierz-Pauli action for spin-2 particle): {\cal L}_{H-E}^{(2)}=-\frac{1}{2}\left[\frac{1}{4}h_{\mu\nu,\rho}^2-\frac{1}{2}h_{\mu}^2+\frac{1}{2}h^{\mu}h_{,\mu}-\frac{1}{4}h_{,\mu}^2\right].

[2] There are many subtleties involving T_{\mu\nu} in GR and you are welcome to read S. Weinberg’s ‘Gravitation & Cosmology’. Also this .pdf file covers Second & First Order formalism in an irritating manner but is quite good nonetheless!

[3] You can do a similar analysis as was done in the previous post for the case of Maxwell’s EM theory for choosing this particular gauge condition.