Pan of Andromeda galaxy

Last year, NASA/ESA released a giant image of the Andromeda galaxy taken by the Hubble Space Telescope. At 4.3 Gb and 1.5 billion pixels this composite of 411 images is completely impractical for most of us, but fortunately one random internet denizen created a stunning panning video:

The Andromeda galaxy is the nearest large galaxy to our own galaxy, the Milky Way. (There are also several dozen smaller galaxies in our “Local Group”). Even though it is 2 million light years away, you can see Andromeda with the naked eye. In the video, the scenery gets brighter towards the end, as the view approaches the galactic centre where there are more stars. NASA has more details, and you can also download the image in various sizes or use a zoomable browser tool.

Virtual particles and the Nobel Prize

The 2016 Nobel Prize in Physics was recently awarded “for theoretical discoveries of topological phase transitions and topological phases of matter“. The following animation shows one aspect of this research:

Vortex (left) and antivortex (right) emerging from the spins of atoms (arrows) in a thin sheet of magnetic material. Credit: Brian Skinner
A vortex-antivortex pair. Credit: Brian Skinner

Picture a thin sheet of magnetic material, with each arrow representing a single atom and the direction of its “spin”. At the lowest energy, all the spins line up in the same direction. Add some energy, and you can get a “vortex” (left) and “antivortex” (right), which exist in a pair, remaining bound together.

But add even more energy and there is a critical level where the vortex and antivortex can separate. This is named the “Kosterlitz-Thouless transition” after two of the Nobel Prize awardees. It is a phase transition, meaning an abrupt change of state like the melting of ice into water at around 0°C or the evaporation of water into steam at around 100°C. (My summary is based on a very readable introduction.)

The vortex and antivortex almost have the appearance of being literal concrete particles moving to the left or right, however it is clear from the animation they are only emergent from patterns of atoms spinning around. There are many examples of such “virtual” or “emergent” particles in physics, which leads us to an intriguing video by MinutePhysics. (Speaking of abrupt transitions!)

The video describes virtual particles such as an electron “hole” which is simply a gap in an otherwise densely packed sea of electrons. It also describes emergent properties such as electrons behaving as if they had very different mass, charge, or spin, in certain circumstances. Hopefully you will enjoy the physics, or in the very least the spinning Lego models.  🙂

The helical model: do planets move in spirals?

A 2012 viral video showed the planets moving in a spiral (“helix”) pattern due to the Sun’s motion through space. It also criticised the “heliocentric” conception of the Sun as being at rest with the planets on roughly circular orbits around it. This raises an interesting question about frames of reference:

(See also the 3rd and improved version embedded later). The author, music producer “DjSadhu”, has made a beautiful animation complete with Tron-style trails for artistic effect. However the main issue is the claim, “The old heliocentric model of our solar system… is not only boring but incorrect.” He continues, “Our Solar System moves through space at 70,000 km/hr”. He calls the planet orbits “rotation” for the stationary Sun perspective, and “vortex” for the moving Sun perspective; this is not standard terminology but we can understand his point.

This issue is that it is equally valid to choose either frame of reference. If we choose a non-rotating frame centred on the Sun, then from this perspective the Sun is at rest and the planets move in circles (approximately). If instead we choose a non-rotating frame centred on our Milky Way galaxy, then from this perspective the Sun is moving at 800,000 km/h (a dozen times higher than the figure in the video) and the planets move in helices, approximately. We could take this further and incorporate the galaxy’s own motion relative to the local universe, or any other natural (described earlier) or hypothetical motion one chooses.

The animator scoured NASA’s website but couldn’t find the helical model. He is probably correct that most of the public has an “incomplete” view, and that “even astronomers” don’t see it this way “even though they may have all the facts that support it.” However, neither would this model be a surprise to them. The concept of relativity of motion is well-known in physics — look up “Galileo’s ship”, a celebrated idea from 400 years ago. I suspect that many physicists would indeed think, “Oh that’s interesting, I hadn’t thought of it that way”, but then also quickly shrug their shoulders and think, “But it’s correct.” But on the other hand, the video fails to understand the merits of the usual conception: it works and it’s simpler! If you are studying planetary orbits in the Solar System, then typically you would ignore external influences as being very minor, and likely choose a coordinate system centred on the Sun (which gives an effective interpretation that the Sun is not moving). The principle of relativity — that the laws of physics are independent (in some sense) of the frame you choose — is a cornerstone of physics, and was furthered by Einstein amongst others. The animator is clearly unaware of what physics/mathematics/philosophy even says on this topic.

Astronomers Phil Plait and Rhys Taylor raised other issues, especially with a second video, including:

  • the Sun does not precede the planets (DjSadhu claims this criticism only applies to the 2nd video), and it is not “dragging the planets in its wake”
  • the Sun does not follow a spiral pattern around the galaxy — this is a misunderstanding of Earth’s precession — but the Sun does bob up and down a little
  • the plane of the Solar System makes an angle of 60° with the Sun’s path through the galaxy, not 90°
  • the correct terminology is “helix”, not “vortex” which applies to fluid flow. The animator’s distinction between “rotation” and “vortex”
  • dubious sources
  • the metaphysical analogy “Life spirals” with pictures of spiral aloe, a fern, rose, spiral galaxy, DNA double helix, shell, and a plughole vortex, was never going to go down well with many scientists.

Taylor wrote:

[Y]ou presented the idea of helical paths as though it were some revolutionary new model. You could have very easily checked with more or less any astronomer who would have told you that we already know this is the case. True, a shiny animation did not exist to show it… [B]ut in context it was saying, “I’m an unqualified DJ who’s overturned all of astronomy“.

To his credit, the animator listened to many of these criticisms. He did also request that people focus on the central claim. Putting aside some things, at his best he writes, “I’m willing to take it down a notch and say there’s more to reality than the heliocentric dinner-plate diagrams. Fair enough?”

This third video, version “2.0”, was praised by Taylor as a “win-win scenario”, stating “bravo, Sadhu, I salute you.” I am discussing this story because I feel it has more merits than flaws overall. So thank-you DjSadhu for sharing your artistic talents! See related animations by Vsauce (16:55–17:54 point, 19:48–end), and Taylor.

Motion of the Milky Way

Our small planet is part of a complicated hierarchy of structure in the heavens:

  • The Earth rotates once per day, so a person standing on the equator moves at 1700 km/h, relative to the centre of the Earth
  • The Earth orbits the Sun at 100,000 km/h, relative to the Sun (in a non-rotating frame of reference)
  • The Sun orbits the centre of our Milky Way galaxy at 800,000 km/h
  • The Milky Way is approaching the centre of our “Local Group” of galaxies at 200,000 km/h. (This is my rough estimate, based merely on the fact that Andromeda and the Milky Way are approaching one another at twice this speed, and these are the dominant two members of the galaxy group.)
  • The Local Group is falling towards the Virgo Cluster at around 400,000 — 1,000,000 km/h, the “Virgocentric flow”. (This is after subtracting the Hubble flow. Note the Local Group and Virgo Cluster are both contained within the Virgo Supercluster, an even larger structure.)
  • The Virgo Supercluster is moving towards the “Great Attractor” region at 1,000,000 km/h, according to an older source. (The Great Attractor is due to the Hydra-Centaurus Supercluster, or the even larger Laniakea Supercluster which encompasses all of the above and more. The Norma Cluster marks the centre.)
  • The Laniakea Supercluster is moving towards the Shapley Supercluster.
Map of the sky showing the "hot" and "cold" spots of the cosmic microwave background (CMB). This unevenness ("anisotropy") is due to the motion of the Solar System, as the Earth's motion relative to the Sun has already been subtracted. This is from the COsmic Background Explorer (COBE) satellite in the early 1990s.
Map of the sky showing the “hot” and “cold” spots of the cosmic microwave background (CMB). This unevenness or dipole is due to the motion of the Solar System, where the Earth’s motion relative to the Sun has presumably already been subtracted. This is from the COsmic Background Explorer (COBE) satellite in the early 1990s, the first detailed map. In most pictures of the CMB this anisotropy has already been subtracted out, leaving much finer hot/cold dimples.

Going back a step, an alternate method is to measure the cosmic microwave background (CMB). This radiation is nearly uniform in all directions, but shows a hot and cold spot (see Lineweaver 1996  for history). Since this is 100 times more pronounced than the finer fluctuations, it makes sense to interpret it as a Doppler effect due to motion. Hence, the Solar System’s motion is calculated as 1,300,000 km/h in the direction of the constellation Leo. By subtracting off the Sun’s estimated motion, the Local Group has a velocity of 2,200,000 km/h in the direction of the constellation Hydra. This is relative to the “CMB rest frame”, assumed to coincide with the Hubble flow, which is the average motion of matter at large scales and is thought of as being “at rest”. However understand this “rest frame” is just a natural and convenient choice, and not the centuries-old concept of “absolute rest” held by Isaac Newton.

Too many talks to remember…

It has been a hectic but successful day, with 12 hours of cycling around Brisbane and attending talks! I started with a part-drive, part-cycle to Southbank, navigating the rain, to watch two documentaries screened for the World Science Festival. “Mapping the Future: The Power of Algorithms” was an interesting discussion of what “predictive analytics” based on “big data” can foretell of human behaviour. “The Joy of Logic” was too introductory for me, but I was interested in the quirky anecdotes about the Vienna Circle of philosophers.

Next I cycled to the University of Queensland to hear Scott Stephens, editor of the (Australian) ABC’s Religion and Ethics website, on “To See or Not to See: Recovering Moral Vision in a Media Saturated Age”. He was critical of the pettiness of media in a democratic society, citing causes including commercialisation of news, the Watergate scandal, and the media’s change from reporting on politics to deliberately influencing it. Also the rise of social media means news organisations pander to popular taste and attempting to “go viral”. I was reminded of Alain de Botton’s commentary and alternative news experiment.

Immediately afterwards I rushed off to a presentation by George Musser, an editor at Scientific American. Researchers feel popular science reporting at this level and below can be too “dumbed-down” and/or sensationalist. Musser tried to unify the roles of “scientist” and “journalist”: science is his original background, but he also defended a journalist perspective to his audience. He said hot topics include cosmology, anything with “quantum” in the title, mind/consciousness, and others (I can certainly see these emphases in the science festival). But trends change — dinosaurs used to be popular, as was water on Mars but people are sick of hearing about that.

The next talk would be a personal highlight. But first I’ll mention for completeness that last night I attended “The first scientists: Aboriginal science in Queensland” panel discussion. The room was completely packed, the most full for the science festival so far, apparently. There were interesting tidbits such as some man in remote northern Queensland who lost part of a finger to a crocodile, then wrapped it in a local bark, a natural anaesthetic; I would have preferred more concrete examples like this.

Killing tensor for Friedmann-LemaƮtre-Robertson-Walker (FLRW) spacetime

The quantity

K_{\mu\nu}\equiv a^2(g_{\mu\nu}+v_\mu v_\nu)

is a Killing tensor field for Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime, which models a homogeneous and isotropic universe. We can use it to conveniently derive the cosmic redshift of photons, or decay of “peculiar velocity” for particles with mass. g_{\mu\nu} is the metric tensor, a=a(t) is the “scale factor”, and v^\mu is the 4-velocity of observers comoving with the Hubble flow with components (1,0,0,0) in the usual coordinate choices.

The number:

K^2\equiv K_{\mu\nu}u^\mu u^\nu

is conserved along a geodesic worldline \mathbf u (more precisely, \mathbf u is the tangent vector under some affine parameter). Such a conserved quantity exists because Killing tensors correspond to symmetries of spacetime.

We can evaluate

K^2=a^2(\mathbf u\cdot\mathbf u+(\mathbf u\cdot\mathbf v)^2)

A photon has norm-squared \mathbf u\cdot\mathbf u=0, and energy E=-\mathbf w\cdot\mathbf b as measured by a Hubble observer (assuming you chose the affine parameter to give the 4-momentum). A massive particle has norm-squared \mathbf u\cdot\mathbf u=-1, and Lorentz factor \gamma=-\mathbf u\cdot\mathbf v as measured by a Hubble observer (assuming the affine parameter is proper time). The usual results follow easily. In my view this the most elegant derivation.

Note you can tweak some quantities, because multiplying a conserved quantity by a constant gives another conserved quantity, and the same applies for a Killing vector/tensor. So we may use the unnormalised scale factor R(t) in place of a(t)\equiv R(t)/R_0, because K'_{\mu\nu}\equiv R_0^2K_{\mu\nu} is also a Killing tensor, with conserved quantity K'=R_0K. As another example, for a timelike particle with constant mass m, we may define K''=mK which is also conserved.

Curiously, the tensor contains the spatial projector h_{\mu\nu}=g_{\mu\nu}+v_\mu v_\nu for the Hubble observers. This is just the “space” part of the metric, along the usual homogeneous and isotropic spatial slices. We can rearrange the conserved quantity to give: h_{\mu\nu}b^\mu b^\nu=K^2/a^2, then take the square root. In words, the length of the spatial part of the tangent vector is inversely proportional to the scale factor. More simply, Hubble observers measure things to lessen? (I need to think about the spatial geodesic case more…) These results sound familiar and unremarkable, but are rigorous and general.

The only textbook I know which identifies it as a Killing tensor is Carroll (2004, §8.5) . (However Wald (1984, §5.3a) deserves credit for using a Killing vector in the same fashion. See my forthcoming post on FLRW Killing vectors.) One source which gives the tensor is Maharaj & Maartens (1987a, §4)  and (1987b, §3) . (While they identify the tensor, and the resulting conserved quantity which they interpret as linear momentum, the term “Killing tensor” is not mentioned.) Also there is similar content in much older sources, including Robertson & Noonan (1968) .

The coordinate choice should not matter, as we can easily write the expression in a tensorial way. One can check that indeed \nabla_{(\alpha}K_{\mu\nu)}=0, where the parentheses denote symmetrisation of the indices.)

(Poster) Static vs Falling: Time slicings of Schwarzschild black holes

I presented this poster, “Static vs Falling: Time slicings of Schwarzschild black holes” at the GR21 conference in New York City, July 2016. You can download a PDF version. A big thank-you to those who gave feedback, especially to Jiří Podolský who encouraged me to publish it! The section on coordinate vectors has been updated. Also there is a major update to the poster.Schwarzschild slicings 40dpi

Tetrad for Schwarzschild metric, in terms of e

The following is a natural choice of orthonormal tetrad for an observer moving radially in Schwarzschild spacetime with “energy per unit rest mass” e:

    \begin{align*} e_{\hat 0}^\mu &= \left(e\Schw^{-1},\pm\eroot,0,0\right) \\ e_{\hat 1}^\mu &= \left(\pm\Schw^{-1}\eroot,e,0,0\right) \\ e_{\hat 2}^\mu &= \left(0,0,\frac{1}{r},0\right) \\ e_{\hat 3}^\mu &= \left(0,0,0,\frac{1}{r\sin\theta}\right) \end{align*}

The components are given in Schwarzschild coordinates. (The ± signs are not independent — they must be either both +1 or both -1. Note that e does not distinguish between inward and outward motion. There is additional freedom to define any of these vectors as their negative.)

We normally think of e as invariant, where there is a presumption of freely falling / geodesic motion, but even if not we can regard it as an instantaneous value.

\evec{0} is the 4-velocity computed previously. The other vectors can be obtained from substituting \gamma=e\Schw^{-1/2} and V=-\frac{1}{e}\sqrt{e^2-1+\frac{2M}{r}} into the tetrad here. \gamma is determined from -\fvec u\cdot\fvec u_{\rm obs}=\gamma and the equation for e above, then V follows from inverting \gamma\equiv(1-V^2)^{-1/2}. This orthonormal frame is useful for determining the object’s perspective, e.g. tidal forces, visual appearances, etc.

Tetrad for Schwarzschild metric

Suppose an observer u moves radially with speed (3-velocity) V relative to “stationary” Schwarzschild observers, where we define V<0 as inward motion. Then one natural choice of orthonormal tetrad is:

    \begin{align*} (\evec{0})^\alpha &= \left(\gamma\Schw^{-1/2},V\gamma\Schw^{1/2},0,0\right) \\ (\evec{1})^\alpha &= \left(V\gamma\Schw^{-1/2},\gamma\Schw^{1/2},0,0\right) \\ (\evec{2})^\alpha &= \left(0,0,\frac{1}{r},0\right) \\ (\evec{3})^\alpha &= \left(0,0,0,\frac{1}{r\sin\theta}\right) \end{align*}

where the components are given in Schwarzschild coordinates. This may be derived as follows.

The Schwarzschild observer has 4-velocity

    \[\fvec u_{\rm obs}=\left(\Schw^{-1/2},0,0,0\right)\]

because the spatial coordinates are fixed, and the t-component follows from normalisation \fvec u_{\rm obs}\cdot\fvec u_{\rm obs}=-1 (Hartle §9.2).

Now the Lorentz factor for the relative speed satisfies -\fvec u\cdot\fvec u_{\rm obs}=\gamma, and together with normalisation \fvec u\cdot\fvec u=-1 and the assumption that the θ and φ components are zero, this yields \evec{0}\equiv\fvec u given above.

We obtain \evec{1} by orthonormality: \evec{1}\cdot\evec{0}=0 and \evec{1}\cdot\evec{1}=1, and again making the assumption the θ and φ components are zero. Note the negative of the r-component is probably an equally natural choice. Then \evec{2} and \evec{3} follow from simply normalising the coordinate vectors.

Strictly speaking this setup only applies for r>2M, because stationary timelike observers cannot exist inside a black hole event horizon! Yet remarkably the formulae can work out anyway (MTW …) .  An alternate approach is local Lorentz boost described shortly.

Hartle … Also check no “twisting” etc…