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How to convert between frame and coordinate bases

This article describes how to transform components of vectors or other tensors between a coordinate basis and an arbitrary frame / tetrad. This process is more general than the transformation between two coordinate bases as found in any introductory general relativity course. Some frames are “non-holonomic” meaning they do not arise from any set of coordinate basis vectors, also there may be situations in which a coordinate representation is inconvenient or not known. I also outline how to implement the transformations in a computer algebra system (CAS).

Effectively we only work in a single tangent space on the manifold, so it turns out to be just a linear algebra problem. My description is based on Carroll textbook (§J) and de Felice & Clarke (§4.2) who assume the frame is orthonormal, however I simply assume it is a basis: that it spans the tangent space and is linearly independent. So suppose we have coordinates $x^\mu$ , and a frame $(\mathbf e_a)$ with components $e_a^{\hphantom a\mu}:=(e_a)^\mu$ in the coordinate system, that is:

$\mathbf e_a = e_a^{\hphantom a\mu}\boldsymbol\partial_\mu$

in terms of coordinate basis vectors. I use Latin indices to specify vectors in the tetrad frame, and add a hat for orthonormal frames. I use Greek indices for coordinate components, for example $(e_0)^\mu$ for the vector $\mathbf e_0$ . (In place of our $e_a^{\hphantom a\mu}$ , de Felice & Clarke write $\lambda_{\hat a}^{\hphantom ai}$ , and Carroll swaps the index order to $e^\mu_{\hphantom\mu a}$ .) In a CAS we can implement the frame as a $4\times 4$ array / matrix called “ $\texttt{frame}$ ” say, reading the indices of $e_a^{\hphantom a\mu}$ from left to right but ignoring their up-or-down placement. This ordering conveniently gives an array of “vectors”:

$\texttt{frame} := \big((e_0^{\hphantom 00},\ldots,e_0^{\hphantom 03}),(\cdots),(\cdots),(\cdots)\big)$

However there is a tradeoff that vectors are placed in rows instead of the more standard column vector representation, because matrix indices refer to the row first and column second. We also define quantities $(e^b_{\hphantom b\mu})$ implemented as a matrix “ $\texttt{dualframe}$ “, which give the coordinate basis vectors in terms of the new frame:

$\partial_\mu = e^b_{\hphantom b\mu}\mathbf e_b$

It follows from linear independence that $e_a^{\hphantom a\mu}e^b_{\hphantom b\mu}=\delta_a^b$ , hence as matrices: $\texttt{dualframe}=(\texttt{frame}^\top)^{-1}$ . The transpose is required because of the index summation order, since the convention for matrix multiplication is $(AB)_{ij}:=\sum A_{ik}B_{kj}$ . This point could easily be missed when references call it “inverse” with more general index summation in mind. Note summing over the Latin indices also returns the identity: $e_a^{\hphantom a\mu}e^a_{\hphantom a\nu}=\delta^\mu_\nu$ .

Now suppose a vector $\mathbf q$ is specified by its coordinate basis components $q^\mu$ , which we implement as a 4-element array $\texttt{Q}$ . Since $\mathbf q=q^\mu\boldsymbol\partial_\mu$ , substituting the previous expression for $\partial_\mu$ and using linear independence gives the components in the new frame (note the Latin index) as: $q^b = e^b_{\hphantom b\mu}q^\mu$ . Programmatically this is the matrix multiplication $\texttt{dualframe*Q}$ , at least for my CAS does not distinguish between a row and column vector but automatically matches the dimensions. Now suppose we have $m$ different vectors, stored in an $m\times 4$ matrix $\texttt{QQ}$ say (typically $m=4$ ). These are processed in a batch operation by converting to column vectors, applying the transformation, then transposing back, so the components are: $(\texttt{dualframe*QQ}^\top)^\top = \texttt{QQ*dualframe}^\top$ , in the new frame.

Now consider the dual bases. In the coordinate dual basis, the vector dual to $\mathbf q$ has components $q_\mu = g_{\mu\nu}q^\nu$ . These components can be implemented as an array $\texttt{dualQ} = \texttt{G*Q}$ where $\texttt{G}$ is the matrix $(g_{\mu\nu})$ and the row / column vector distinction is ignored as before. Again we can lower multiple vectors in one step via $(\texttt{G*QQ}^\top)^\top = \texttt{QQ*G}$ .

The dual to the new frame satisfies $\mathbf e^b(\mathbf e_a) = \delta^b_a$ by definition, hence

$\mathbf e^b = e^b_{\hphantom b\mu}dx^\mu$

which may be validated by substitution, and these components are just $\texttt{dualframe}$ again. Similarly

$dx^\mu = e_a^{\hphantom a\mu}\mathbf e^a$

which are the components $\texttt{frame}$ again. Carroll’s description for an orthonormal frame is true for any frame:

The vielbeins [ $e^b_{\hphantom b\mu}$ ] thus serve double duty as the components of the coordinate basis vectors in terms of the orthonormal basis vectors, and as components of the orthonormal basis one-forms in terms of the coordinate basis one-forms; while the inverse vielbeins serve as the components of the orthonormal basis vectors in terms of the coordinate basis, and as components of the coordinate basis one-forms in terms of the orthonormal basis.

Likewise Schutz’ textbook (§3.3) description of Lorentz transformations holds more generally:

…components of one-forms transform in exactly the same manner as basis vectors and in the opposite manner to components of vectors.
[…Whereas basis one-forms transform] the same as for components of a vector, and opposite that for components of a one-form.

We may also define (de Felice & Clarke, eqn. 4.2.5):

$e_{a\mu} := \mathbf e_a \cdot \partial_\mu$

which I interpret as a definition. This evaluates to $g_{\mu\nu}e_a^{\hphantom a\nu$ (or $\texttt{frame*G}$ ), hence $g^{\mu\nu}e_{a\nu} = e_a^{\hphantom a\mu}$ . Define also

$e^{b\nu} := \mathbf e^b \cdot dx^\nu$

which evaluates to $g^{\mu\nu}e^b_{\hphantom b\mu}$ (or $\texttt{dualframe*Ginv}$ , where $\texttt{Ginv}$ is the matrix $(g^{\mu\nu})$ ), hence $g_{\mu\nu}e^{b\nu} = e^b_{\hphantom b\mu}$ . Thus Greek indices are raised and lowered in the familiar way — using the metric components in the coordinate basis). On the other hand the metric components in the new frame are

$g_{ab} := \mathbf e_a\cdot\mathbf e_b = g_{\mu\nu}e_a^{\hphantom a\mu}e_b^{\hphantom b\nu}$

which can be implemented as $\texttt{Gframe} := \texttt{frame*G*frame}^\top$ . In the particular case of an orthonormal frame $g_{\hat a\hat b}=\eta_{\hat a\hat b}$ , so in this case Latin indices are raised and lowered with the Minkowski metric. The metric in the dual frame is

$g^{ab} := \mathbf e^a\cdot\mathbf e^b = g^{\mu\nu}e^a_{\hphantom a\mu}e^b_{\hphantom b\nu}$

so define $\texttt{Gdualframe} := \texttt{dualframe*Ginv*dualframe}^\top$ . These are matrix inverses: $g_{ab}g^{bc}=\delta_a^c$ . We can show Latin indices are raised or lowered using this frame metric, so for example $e_b^{\hphantom b\mu} = g_{ab}e^{a\mu}$ .

With all these definitions of components as metric inner products between quantities, we may wonder if the original frame components can also be expressed this way. Indeed they can: $e_a^{\hphantom a\mu} = \mathbf e_a(dx^\mu)$ and $e^a_{\hphantom b\mu} = \mathbf e^a(\partial_\mu)$ , where the vectors and dual vectors are acted on one another. The metric is implicit in the summation, because as a (1,1)-tensor it is just the identity. But the (1,1)-tensor $e_b^{\hphantom b\nu}\mathbf e^b\otimes\partial_\nu$ made from the “frame” components is also just the identity (see Carroll), so it and the metric tensor are equal. Input $\mathbf e_a$ and $dx^\mu$ into this tensor and it indeed returns $e_a^{\hphantom a\mu}$ . We can do similarly with the dual frame.

For higher rank tensors, their components are expressed in the new frame as e.g. (de Felice & Clarke, Hartle textbook §20.3, §21.2)

$T^{ab}_{\hphantom{ab}cd} = T^{\mu\nu}_{\hphantom{\mu\nu}\sigma\tau} e^a_{\hphantom a\mu} e^b_{\hphantom b\nu} e_c^{\hphantom c\sigma} e_d^{\hphantom d\tau}$

My CAS multiplies higher rank “matrices” $\texttt{A*B}$ by contracting the last index of $\texttt{A}$ with the first index of $\texttt{B}$ . Hence we can only change two indices of $T$ by this method, short of reordering the indices halfway through. There is another inbuilt method “TensorContract” which I will relate sometime later. Of course you could just program in the sum manually, but I am seeking an elegant solution for aesthetic satisfaction, also because inbuilt operations are probably more optimised. Finally you can continue to mix Greek and Latin (coordinate and frame) indices, see Carroll and I will add an example later.

Fully covariant force in general relativity

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 textbook , §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 textbook §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 textbook §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.

Time slicings of black holes poster

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.

Research – Colin MacLaurin

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, “Expanding space, redshifts and rigidity: Conceptual issues in cosmology“. My Master’s thesis in general relativity.

2015 Master’s thesis

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”.