Local Lorentz boost in coordinate-independent notation

Difficulty:   ★★★☆☆   undergraduate

The Lorentz boost between two reference frames can be expressed as a (1,1)-tensor \boldsymbol\Lambda, interpreted as an operator on vectors. Here we re-express this well known fact using a general, index-free, coordinate-independent, 4-vector notation, which is valid locally in curved spacetime.

Recall the prototypical Lorentz boost on Minkowski spacetime:

    \[\Lambda^\mu_{\hphantom\mu\nu} = \begin{pmatrix} \gamma & -\beta\gamma & 0 & 0 \\ -\beta\gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}.\]

This is a boost in the x-direction by speed \beta or Lorentz factor \gamma = (1-\beta^2)^{-1/2}. It maps an arbitrary vector X^\mu to \Lambda^\mu_{\hphantom\mu\nu}X^\nu. Numerous authors generalise to arbitrary boost directions, such as Møller  §18; MTW  §2.9; or Tsamparlis  §1.7. This typically involves separate transformations of time and 3-dimensional space: t' = \gamma(t - \vec n\cdot\vec r) and \vec r' = \vec r + (\frac{\gamma-1}{\beta^2}\vec r\cdot\vec n - \gamma t)\vec n. The arrows signify 3-dimensional vectors, \vec r is the position in 3-space, and \vec n is the relative 3-velocity. The space part uses beautiful, coordinate-independent vector language. However the time part requires privileged coordinates adapted to the observers. We will derive a 4-vector analogue.

Consider two 4-velocity vectors \mathbf u and \mathbf n (located at the same point, if in curved spacetime). They are related by the Lorentz boost:

    \[\mathbf n = \gamma(\mathbf u+\beta\hat{\vec{\mathbf n}}) = \gamma(\mathbf u+\vec{\mathbf n}),\]

where \gamma = -\langle\mathbf u,\mathbf n\rangle, the unit vector \hat{\vec{\mathbf n}} points in the boost direction, and \vec{\mathbf n} = \beta\hat{\vec{\mathbf n}} is the relative velocity. This is the 4-vector analogue of the familiar coordinate boost t' = \gamma(t-\beta x). Combined with the space boost given shortly, this forms a local Lorentz transformation. While the plus sign makes the above appear an inverse boost, this is only because vectors (as whole entities) transform inversely to coordinates. Rearranging:

    \[\vec{\mathbf n} = \gamma^{-1}\mathbf n - \mathbf u.\]

This is the relative velocity of the observer \mathbf n as determined in \mathbf u’s frame, as explained previously. It is equivalent to the \vec n introduced in the 3-dimensional spatial transformation, except now treated as a 4-vector. It is orthogonal to \mathbf u, with length \beta. Conversely, the relative velocity of \mathbf u as determined in \mathbf n’s frame is \vec{\mathbf u} = \gamma^{-1}\mathbf u - \mathbf n. Now, the vector analogue of the usual boosted spatial coordinate x' = \gamma(x-\beta t) is \gamma(\hat{\vec{\mathbf n}}+\beta\mathbf u). After multiplying by \beta:

    \[\vec{\mathbf n} \mapsto \gamma(\vec{\mathbf n} + \beta^2\mathbf u) = \mathbf n - \gamma^{-1}\mathbf u = -\vec{\mathbf u}.\]

Hence the relative velocity vectors are boosted into one-another, aside from a minus sign (Jantzen+ 1992  §4). This generalises the Newtonian result \vec n = -\vec u. So we have the boost’s action on the orthogonal vectors \mathbf u and \vec{\mathbf n}, plus it is the identity on the 2-dimensional spatial plane orthogonal to both, hence:

    \[\begin{aligned} \Lambda^\mu_{\hphantom\mu\nu} &= (g^\mu_{\hphantom\mu\nu} + u^\mu u_\nu -\beta^{-2}\vec n^\mu \vec n_\nu) - n^\mu u_\nu -\beta^{-2}\vec u^\mu \vec n_\nu \\ &= g^\mu_{\hphantom\mu\nu} + (u^\mu-n^\mu)u_\nu + \frac{\gamma}{\gamma+1}(u^\mu + n^\mu)\vec n_\nu, \end{aligned}\]

using \vec{\mathbf u}+\vec{\mathbf n} = -(1-\gamma^{-1})(\mathbf u+\mathbf n) and (\gamma-1)/\beta^2\gamma = \gamma/(\gamma+1). It is a good exercise to check the contractions with u^\nu, \vec n^\nu, or any X^\nu orthogonal to both. In index-free notation,

    \[\boldsymbol\Lambda = \mathbf g^{\sharp\flat} + (\mathbf u-\mathbf n)\otimes\mathbf u^\flat +\frac{\gamma}{\gamma+1}(\mathbf u+\mathbf n)\otimes\vec{\mathbf n}^\flat.\]

The “flat” symbol just means: lower an index. Equivalently, in terms of the initial observer and boost velocity alone:

    \[\boldsymbol\Lambda = \mathbf g^{\sharp\flat} + \big((1-\gamma)\mathbf u - \gamma\vec{\mathbf n})\otimes\mathbf u^\flat + \big(\gamma\mathbf u + \frac{\gamma^2}{\gamma+1}\vec{\mathbf n}\big)\otimes\vec{\mathbf n}^\flat,\]

in which case the relative speed may be obtained from \beta^2 = \langle\vec{\mathbf n},\vec{\mathbf n}\rangle. This is equivalent to MTW’s Exercise 2.7 which uses Cartesian coordinates, after adjusting various minus signs because I use vectors not vector components. In terms of the 4-velocities alone, we have the curiously symmetric expression:

    \[\boldsymbol\Lambda = \mathbf g^{\sharp\flat} + \frac{1}{\gamma+1} (\mathbf u+\mathbf n)\otimes(\mathbf u+\mathbf n)^\flat - 2\mathbf n\otimes\mathbf u^\flat.\]

Formulae are useful machines, allowing you to blithely turn the handle to crank out a result. This contrasts with my usual emphasis on conceptual understanding, and drawing a picture (at least mentally). However Lorentz boosts have many counter-intuitive or seemingly paradoxical effects. It is easier to make a mistake if you reason from first-principles alone. Of course the algebra does originate from careful thinking about foundations, and having multiple approaches is a check of consistency.

Boosts are paramount for comparing physical quantities between frames. Some textbooks present the general Lorentz boost in Minkowski spacetime with Cartesian coordinates. Our abstract vector formulation allows direct application to local boosts in arbitrary spacetime, such as Kerr or FLRW, in any coordinate system. I don’t remember seeing the formulae here in the literature, but someone should have done it somewhere. The Jantzen+ paper was an inspiration, and the same authors define various further quantities (projections, in fact) in Bini+ 1995 .

🡐 relative velocity | relative kinematics | ⸻ 🠦

Leave a Reply

Your email address will not be published. Required fields are marked *