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

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The quantity

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

is a Killing tensor field for Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime which models a homogeneous and isotropic universe. (\mathbf g is the metric tensor, \mathbf w is the 4-velocity of observers comoving with the Hubble flow, and a=a(t) is the scale factor. 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.) This is useful because Killing vectors and tensors correspond to symmetries of spacetime and hence conserved quantities. We can use K_{\mu\nu} to conveniently derive the cosmic redshift of photons, and decay of “peculiar velocity” for particles with mass. We start with the quantity:

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

which is conserved along a geodesic (more precisely, \mathbf b is the tangent vector to the geodesic, under some affine parameter e.g. proper time). It follows

K^2=a^2(\mathbf b\cdot\mathbf b+(\mathbf w\cdot\mathbf b)^2)

For a photon, its tangent vector has norm-squared \mathbf b\cdot\mathbf b=0, and energy -\mathbf w\cdot\mathbf b as measured by a Hubble observer (or proportional to this, depending on your choice of affine parameter). For a massive particle, its norm-squared is \mathbf b\cdot\mathbf b=-1, and -\mathbf w\cdot\mathbf b=\gamma which is the Lorentz factor for the motion relative to the Hubble flow observer (assuming its worldline is parametrised by proper time). The usual results follow. In my view this is more elegant than most textbook approaches.

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 replace a(t)=R(t)/R_0 by the unnormalised scale factor R(t), because R_0^2K_{\mu\nu} is also a Killing tensor, with conserved quantity K'=R_0K. Alternatively 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}+w_\mu w_\nu for the Hubble observers. This is just the spatial part of the metric, along the usual homogeneous and isotropic 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 spatial length of the tangent vector to a geodesic is inversely proportional to the scale factor. More precisely: this assumes an affine parameter, also the length is determined by Hubble flow observers. More simply, Hubble observers measure things to lessen? (I need to think about the spatial geodesic case more…)

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) .

Research – Colin MacLaurin

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My research area is general relativity. These papers are drafts not yet ready for arXiv, but exhibit my work prior to Europe conferences.  — Colin MacLaurin

  • 2017, “Distance in Schwarzschild spacetime” (draft). Observers with “energy per mass” e measure a radial distance |e|^{-1}dr. I overview four different tools to measure spatial distance — spatial projector, tetrads, adapted coordinates, and radar — which are locally equivalent. Though spatial distance is foundational, it remains underdeveloped. I clarify subtleties, and counteract the Newton-esque over-reliance on the static distance (1-2M/r)^{-1/2}dr.
  • 2017, “Cosmic cable” (draft). A cosmic-length cable could be used to mine energy from the expansion of the universe. Beyond sci-fi, this is instructive for relativity pedagogy. The dynamics include motion-dependent distance, and time-dilation which reduces the force, effects which are missed in most existing treatments.

2015 Master’s thesis

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Here is my Master of Science thesis, titled “Expanding space, redshifts, and rigidity: Conceptual issues in cosmology“. It was submitted in mid-2015 and supervised by Prof. Tamara Davis at the University of Queensland. I am planning to edit it and write a new foreword, but maybe it is too rugged for arXiv. Still, several papers inspired by it are in production.

I am expanding the material in §7 into a paper on “Measuring distances in Schwarzschild spacetime”. I am also expanding the kinematics of a moving rigid cable (§9, §11) to include force, tension, and power, and apply it to a cosmology spacetime. Existing treatments of both topics typically have “Newtonian” misconceptions but my work properly includes the relativity of distance and simultaneity for instance.

The thesis has a detailed introduction to distance measurement including the spatial projector and “proper metric” (aka “pullback” onto a material manifold) (§3), along with a defense of ruler distance (§6). There is also a detailed introduction to Rindler’s accelerated coordinates (§2.7, §3 etc), followed by a generalising procedure (§8). Also present is an overview of Newtonian cosmology and the Milne model (§4). A major theme is that cosmic redshifts can be variously taken as Doppler, gravitational, cosmic, or a combination of these, but most interpretations aren’t “natural”.

Relativity wiki

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I have decided to start a relativity “wiki”, which is closer to my aims than a blog. Besides, I have experience of making over 16,000 edits on Wikipedia itself. A wiki will allow for better structuring and linking of content, for instance of niche content such as: Black hole → Schwarzschild → Geodesics → “drips” → exact integral. The content is still being polished, and I have many notes which are not integrated yet.

Pan of Andromeda galaxy

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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

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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.  :)

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

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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

The helical model: do planets move in spirals?

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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

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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.