## (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.

## 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.

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…

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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.

## Lunch with some visitors

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This is an epic week for science in Brisbane! Lots of experts are in town for the World Science Festival. The astrophysics group from my uni, the University of Queensland, had lunch today with Josh Frieman (from Chicago) and Douglass Scott (from Vancouver). I had a great time.

Frieman is the director of the Dark Energy Survey (DES), a large international collaboration which is mapping the skies to measure the expansion of the universe by dark energy. Probably every local at lunch except me was a member of either the Australian version “AusDES”, and/or the Australian astrophysics organisation “CAASTRO” which also shouted lunch incidentally. (My research in relativity is a little different from these groups, being more theoretical, but they are the closest I have to a research community here, and so I was very happy to be invited! A big thank-you to Tamara who is conscientious about that.)

I had a good chat with Douglas, whom I was sitting next to. He shared with the table about one of his students who is studying modified gravity theories (that is, other than Einstein’s general relativity), and how every 3 months he knocks on Douglas’ door with another discovery about why general relativity is superior. He generously asked me about my research, and I explained my analysis of distances in relativity, though it might sound trivial. I also mentioned black hole volumes and new coordinate system(s) I had discovered, and that just recently I had been learning about rotating black holes to potentially extend these results to that situation, which I believe no-one has done before apart from a few specific cases.

While Josh (Frieman) is in town for the science festival, Douglas is in Australia to visit a couple of universities. It also turned out he’s going to a music festival tonight, and I joked that his real ulterior motive had come out! He admitted he once gave a talk at Caltech partly because his favourite band was playing in Los Angeles. Tamara left for the science festival. I had a haircut this morning, thinking I should clean up in case I met some of the celebrity physicists in town but, as you can see from the top photo, a shave was beyond me at this point.

## Gravitational waves detected!

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Physicists are very excited, because the first ever direct detection of gravitational waves has just been announced! The signal matches the prediction for two black holes colliding.  This will likely mean a Nobel Prize for someone. This is a tremendous scientific achievement, representing a vast global collaboration between scientists, advanced technology, government funding, and simple good luck.

The signal lasted for just 1/5 of a second, but scientists have extracted an impressive amount of information from it. This video plays the “chirp” which was detected, converting the gravitational wave signal to sound so you can hear it. The video repeats the chirp 8 times, half of those scaled to a higher frequency where human hearing is more sensitive.

But understand that calling it “sound” is metaphorical, for instance when someone gave a demonstration by playing a cello on Australian Broadcasting Corporation (ABC) TV. A gravitational wave is a ripple through the “fabric” of space itself and travels at the speed of light, whereas a sound wave transmits via air molecules bumping together and travels a million times more slowly. It should also be clarified that the gravitational waves would have been emitted for a far longer period than 0.2 seconds, it’s just they were too weak to be detected by us.

Gravitational waves are a consequence of general relativity, and were first predicted by Einstein in 1916. Though not an area of my research so far, I have looked in-depth at the measurement of distances in relativity, which is somewhat related. I look forward to learning and sharing more.

## A new 9th planet?

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There’s news today that some scientists predict the existence of a ninth planet. No one has actually found anything, but this is inferred from the orbits of certain icy objects in the outer Solar System. It may have 10 times the mass of the Earth, and take 10,000 or 20,000 years to orbit the Sun, due to its distance far beyond the known planets.Scientists from the California Institute of Technology (Caltech) claim “Planet Nine” would have an orbit like the above (yellow) to account for the depicted handful of bodies (purple orbits) lying in one direction. They hope to detect it in the next 5 years. It’s not my area, and I have no opinion on this personally, but am happy to wait and see what consensus forms. Still, it’s an opportunity to share some history of planetary discovery.

This has happened before. Neptune was discovered because of irregularities in the orbit of Uranus. Pluto was discovered by the same motivation. (Further irregularities in Neptune and Uranus’ orbits had led to a search. However it turned out Neptune’s mass had been overestimated, and besides Pluto was too small to affect these planets much.) Similarly the unexplained rotation of Mercury’s orbit led to speculation of a new innermost planet “Vulcan”, but just like the Star Trek planet it remains fictional. In fact Einstein successfully explained Mercury’s behaviour using an early version of general relativity.

## World Science Festival in Brisbane

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The World Science Festival is coming to Brisbane on 9–13 March, 2016. This might be the first time it’s been held outside of New York City.

There’s a lot of astrophysics and relativity, including some big names, but it’s expensive. Sean Carroll, author of a good relativity textbook , will discuss the accelerating universe with Nobel prizewinner and Aussie Brian Schmidt (who will also be on “Breakfast with the Brians”), and others. Tamara Davis, my Master’s thesis supervisor, will discuss relativity with string theorist and author Brian Greene and others. The drama “Light Falls” written by Brian Greene about Einstein’s discovery of general relativity looks great, but I don’t want to pay $69/$89. Another drama about Einstein’s personal side was written by Alan Alda, who was the main character of M*A*S*H.

Madness Redefined” also looks interesting. But I think I’ll just drop in briefly on the Sunday, for the free documentary Science and Islam and maybe the free “Street Science” demonstration. I’ve also booked tickets for a maths documentary on “predictive analytics” about modelling our lives, maybe from internet data.

## Tetrad for Schwarzschild metric, in terms of e

Difficulty level:   ★ ★ ★

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

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.

is the 4-velocity computed previously. The other vectors can be obtained from substituting and into the tetrad here. is determined from and the equation for e above, then V follows from inverting . This orthonormal frame is useful for determining the object’s perspective, e.g. tidal forces, visual appearances, etc.

Difficulty level:   ★ ★ ★ ★

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

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

The Schwarzschild observer has 4-velocity

because the spatial coordinates are fixed, and the t-component follows from normalisation (Hartle §9.2).

Now the Lorentz factor for the relative speed satisfies , and together with normalisation and the assumption that the θ and φ components are zero, this yields given above.

We obtain by orthonormality: and , 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 and follow from simply normalising the coordinate vectors.

Strictly speaking this setup only applies for , 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…