Orthonormal tetrad frame for circular motion in Schwarschild spacetime

Difficulty level:   ★ ★ ★

Suppose a test particle undergoes circular motion in Schwarzschild spacetime, and not necessarily at the freefall rate. We assume its worldline is “centred on r=0” so to speak, or to be technical we could speak of the integral of a Killing vector field which is timelike at the given location: this also includes the case of no rotation at all but that’s fine. For simplicity, orient Schwarzschild-Droste coordinates so the motion coincides with the “equator” \theta=\pi/2, and also with increasing \phi-coordinate. Note the r-coordinate is constant. Define the angular velocity as the coordinate ratio

    \[\Omega := \frac{d\phi}{dt} > 0\]

assumed to be constant. We can solve for the 4-velocity of the particle:

    \[u^\mu = \frac{1}{\sqrt{1-2M/r-r^2\Omega^2}}\big(1, 0, 0, \Omega\big)\]

which by definition is also the “time” vector \mathbf{e_{\hat 0}} of the particle’s orthonormal frame. We must have

    \[\Omega<\frac{1}{r}\sqrt{1-2M/r}\]

for the motion to remain timelike. Note that while freefall circular motion requires r>3M to be outside the photon sphere, for accelerated circular motion any r>2M is permissible.

Now for the “space” vectors, it is natural, given our orientation of coordinates, to define two of them as simply normalised coordinate vectors:

    \[(e_{\hat 1})^\mu := \big(0,\sqrt{1-2M/r},0,0\big)\]

    \[(e_{\hat 2})^\mu := \big(0,0,1/r,0\big)\]

This fixes the final vector \mathbf{e_{\hat 3}} in the tetrad up to orientation (overall +/- factor), so we choose the orientation with positive \phi-component:

    \[\frac{1}{\sqrt{1-2M/r-r^2\Omega^2}} \big(r\Omega/\sqrt{1-2M/r},0,0,\sqrt{1-2M/r}/r\big)\]

Then \mathbf{e_{\hat\alpha}}\cdot\mathbf{e_{\hat\beta}}=\pm\delta_{\alpha,\beta} as required, where \delta is the Kronecker delta function. The particle’s 4-acceleration is given by the covariant derivative \nabla_{\mathbf u}\mathbf u. I omit the result, but only the r-component is nonzero, which is not unexpected. The length of the 4-acceleration vector is the magnitude of proper acceleration:

    \[\frac{\lvert M-r^3\Omega^2\rvert \sqrt{1-2M/r}}{r^2(1-2M/r-r^2\Omega^2)}\]

For the special case of geodesic motion the acceleration must vanish, so \Omega=\sqrt{M/r^3}, and the factor 1-2M/r-r^2\Omega^2 reduces to 1-3M/r. See Hartle ( §9.3) for a very different derivation of this. For \Omega=0 the expression reduces to the familiar acceleration of a static observer.

Fully covariant force in general relativity

Difficulty level:   ★ ★ ★

It is often said Newton was fortunate to define force on a particle as the change in momentum f:=dp/dt, not from the change in velocity m\,dv/dt, because the former generalises better. Here the momentum is p:=mv, and clearly the expressions for force coincide if the mass m is constant.

In relativity, the force (4-force) on a particle is usually defined as the change in 4-momentum over proper time as follows:

    \[\mathbf f := \frac{d\mathbf p}{d\tau}\qquad\qquad\textrm{(LIF)}\]

However this expression is only valid in a local inertial frame (LIF), as Hartle (2003 , §20.4) clearly qualifies. Recall, the 4-momentum is \mathbf p := m\mathbf u where \mathbf u is the 4-velocity of the particle. We can split the force into two orthogonal vectors:

    \[\mathbf f = \frac{d(m\mathbf u)}{d\tau} = m\frac{d\mathbf u}{d\tau} + \frac{dm}{d\tau}\mathbf u = m\mathbf a + \frac{dm}{d\tau}\mathbf u\]

where \mathbf a:=d\mathbf u/d\tau (LIF) is the 4-acceleration. The m\mathbf a term is called a “pure force”, because they “create motion in three-dimensional space and correspond to the Newtonian forces”, as Tsamparlis (2010 §11.2) describes, meaning motion in the instantaneous 3-space orthogonal to \mathbf u. The term containing \mathbf u is called a “thermal force”, at least by Tsamparlis. Examples which are at least partly thermal include a particle heated by an external source, or a rocket losing mass. Another example, considered by Einstein apparently, is an object which absorbs two photons with equal energies and opposite directions in the object’s frame, which results in a thermal force but no pure force. On relativistic force, see also Gourgoulhon (2013 §9.5). (Note if the mass does change over time, this is nothing to do with the old-fashioned “relativistic mass” m\gamma dependent on the Lorentz factor, rather we use the modern meaning of mass as “rest mass”.)

Now textbooks and webpages on relativistic mechanics typically assume special relativity, in particular inertial frames within Minkowski spacetime. So how should we generalise to arbitrary coordinates and curved spacetime? According to Hartle (§20.4), the derivative d/d\tau (LIF) generalises to the covariant derivative \nabla_{\mathbf u}. Hence, the fully covariant expression for 4-force is:

    \[\boxed{\mathbf f := \nabla_{\mathbf u}\mathbf p}\]

In words, this is the change of 4-momentum in the direction of the 4-velocity. But in the particle’s frame, its 4-velocity is precisely the “time” direction. So, we could say force is the change of momentum with time in the particle’s frame. So while the mathematics is more general, the concept has clear lineage from special relativity and even Newton!

Now we can repeat the above splitting:

    \[\mathbf f = \nabla_{\mathbf u}(m\mathbf u) = m\nabla_{\mathbf u}\mathbf u + (\nabla_{\mathbf u}m)\mathbf u  = m\mathbf a + \frac{dm}{d\tau}\mathbf u\]

since \mathbf a := \nabla_{\mathbf u}\mathbf u is the usual fully covariant expression, and \nabla_{\mathbf u} of a scalar is d\cdot/d\tau. This expression for force is the same as the specific LIF case above.

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

Difficulty level:   ★ ★ ★ ★

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.)

Time slicings of black holes poster

Difficulty level:   

Below is a copy of my poster “Time slicings of black holes”. It contrasts two different perspectives on Schwarzschild spacetime: by falling and static observers. More technically, I give a family of spacelike foliations which are orthogonal to the worldlines of observers freely-falling radially, and examine the resulting 3-spaces and simultaneity. These properties are contrasted with the static slicing described by the Schwarzschild coordinate t=const. My work is a reaction against the over-emphasis on the static slicing, which has led to many persistent misconceptions, whereas I emphasise space and time are relative. (Of course the 4-dimensional spacetime is independent of the slicing.)

The original version was presented at the general relativity conference GR21 in New York City, 2016, and subsequently other conferences. Below is the 2017 updated version, first presented at the quantum gravity conference “Probing the spacetime fabric” in Trieste, Italy, 2017. [Brief brag moment: luminaries who have viewed and discussed it with me include Jiří Podolský at GR21, and Piotr Chruściel at the “Between Geometry and Relativity” program in Vienna, Austria, 2017.]

A PNG image version is shown below, you can also access a PDF version or even the original.

black holes poster 2017

Research – Colin MacLaurin

Difficulty level:   

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” (edit: removed until ready for arXiv). 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

Difficulty level:   

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”.

Pan of Andromeda galaxy

Difficulty level:   

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

Difficulty level:   

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

Difficulty level:   

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?

Difficulty level:   

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.