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.

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” (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

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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.