Suppose a test particle undergoes circular motion in Schwarzschild spacetime, centred on (loosely speaking) but not necessarily freely-falling. For simplicity, orient Schwarzschild-Droste coordinates so the motion coincides with the “equator” , and also with increasing -coordinate. Note the -coordinate is constant. Define the angular velocity as the coordinate ratio:
We can solve for the 4-velocity of the particle:
which by definition is also the “time” vector of the particle’s orthonormal frame. We must have
for the motion to remain timelike. Note that while freefall circular motion requires to be outside the photon sphere, for accelerated circular motion any 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:
This fixes the final vector in the tetrad up to orientation (overall factor), so we choose the orientation with positive -component:
Then as required, where is the Kronecker delta function. The particle’s 4-acceleration is given by the covariant derivative . I omit the result, but only the -component is nonzero, which is not unexpected. The length of the 4-acceleration vector is the magnitude of proper acceleration:
For the special case of geodesic motion the acceleration must vanish, so , and the factor reduces to . See Hartle ( §9.3) for a very different derivation of this.
It is often said Newton was fortunate to define force on a particle as the change in momentum , not from the change in velocity , because the former generalises better. Here the momentum is , and clearly the expressions for force coincide if the mass 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:
However this expression is only valid in a local inertial frame (LIF), as Hartle (2003 , §20.4) clearly qualifies. Recall, the 4-momentum is where is the 4-velocity of the particle. We can split the force into two orthogonal vectors:
where (LIF) is the 4-acceleration. The 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 . The term containing 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” 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 (LIF) generalises to the covariant derivative . Hence, the fully covariant expression for 4-force is:
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:
since is the usual fully covariant expression, and of a scalar is . This expression for force is the same as the specific LIF case above.
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. is the metric tensor, is the “scale factor”, and is the 4-velocity of observers comoving with the Hubble flow with components in the usual coordinate choices.
is conserved along a geodesic worldline (more precisely, is the tangent vector under some affine parameter). Such a conserved quantity exists because Killing tensors correspond to symmetries of spacetime.
We can evaluate
A photon has norm-squared , and energy as measured by a Hubble observer (assuming you chose the affine parameter to give the 4-momentum). A massive particle has norm-squared , and Lorentz factor 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 in place of , because is also a Killing tensor, with conserved quantity . As another example, for a timelike particle with constant mass , we may define which is also conserved.
Curiously, the tensor contains the spatial projector 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: , 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 , where the parentheses denote symmetrisation of the indices.)
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” measure a radial distance . 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 .
- 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, “Expanding space, redshifts and rigidity: Conceptual issues in cosmology“. My Master’s thesis in general relativity.
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”.
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.
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.
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:
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.
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.