Cosmic Cable poster

Difficulty level:   ★ ★ ★

Here is my recently completed “Cosmic Cable” poster. You can also download a PDF version. Its first appearance is at the GR22 conference in Valencia, Spain, this week.

This work investigates the mechanics of a rigid 1-dimensional object in an arbitrary static, spherically symmetric spacetime. Others have applied such ropes / cables / strings to thermodynamics of black holes (for example scooping up Hawking radiation in a box), or to harvest energy in an expanding universe. It is an interesting exercise in topics which don’t receive a lot of attention, such as extended rigid objects in relativity.

I review the case of static cables, which show a fascinating “redshift” of force effect. I then generalise to a simple case of moving cables, solve for the kinematics, tension, and also the power than can be generated (loosely, this is from a loss of gravitational potential). In case this stuff sounds simple, it is not at all obvious, indeed many papers fail already at the kinematics step. The frame dependence of the quantities is a conceptual challenge.

If you’d like more details I have a proceedings (forthcoming) from the Marcel Grossmann conference of 2018. I will expand this into a longer paper. I would also like to write a pedagogical paper explaining Gibbons (1972)  — his 1-page paper is beautifully concise, yet is hard to understand, has an error (as others have pointed out), and lots of typos.

Difference between special and general relativity?

Difficulty level:   ★ ★ ★

What is the distinction between special relativity (SR) and general relativity (GR)?

It is sometimes said SR can only handle inertial frames, but enough commentators call this a misconception that I must go along with them. A pedagogical paper  on the arXiv today is one example. Also Carroll (2004, §1.2)  writes,

The notion of acceleration in special relativity has a bad reputation, for no good reason. Of course we were careful, in setting up inertial coordinates, to make sure that particles at rest in such coordinates are unaccelerated. However, once we’ve set up such coordinates, we are free to consider any sort of trajectories for physical particles, whether accelerated or not.

This seems a good definition to me: SR is the use of Minkowski coordinates in Minkowski spacetime. You can describe acceleration, but only from within an inertial frame. For example the classic SR textbook Taylor & Wheeler (1992, §2.4)  states, “special relativity is limited to free-float frames”. But from within such frames, they do analyse accelerating particles, see e.g. §3.2. Similarly Misner, Thorne & Wheeler (1973)  even title their section §6.1, “Accelerated observers can be analyzed using special relativity”.

Another definition could be: SR is what you learn in an SR course. In high school I learned about the Lorentz factor, Lorentz transformations, length-contraction, time-dilation, and composition of boosts in the same spatial direction. Undergraduate SR courses presumably have more content, but the term “SR” would still exclude more advanced material, such as Christoffel symbols perhaps, under this definition.

However some textbooks disagree. Misner, Thorne & Wheeler have a solid presentation of 1-forms (§2), and include Fermi-Walker transported tetrads (§6), both in an “SR” context. Gourgoulhon (2013)  takes it to a whole other level, including self-described “rather advanced topics”. He allows not only arbitrary coordinates but non-coordinate bases (§15.4.3), after all the textbook is titled, “Special relativity in general frames”. Gourgoulhon discusses the stress-energy tensor (§19), relativistic hydrodynamics (§21), and even gravitation via historical scalar field theories on flat spacetime (§22). (Of course the stress-energy tensor doesn’t couple to spacetime curvature in this context, so the Einstein field equations are not satisfied.) Personally I would call all this “Minkowski spacetime” rather than “special relativity”! Then again, it could be a publisher’s decision for marketing purposes.

Finally, another definition of SR would be historical, limited to the scope of early papers including Einstein (1905)  and by Minkowski.

In conclusion, I am happy with the definition of SR as Minkowski spacetime using only global inertial frames. Minkowski coordinates would certainly included, with the metric -dt^2+dx^2+dy^2+dz^2, and even simple alternatives such as spherical coordinates -dt^2+dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2), so long as covariant derivatives are not required for a given context. Another time I will discuss the application of SR results in global inertial frames to local orthonormal frames in GR.