Duality of a basis is not duality of individual vectors

Suppose we have a coordinate system (x^\alpha). This defines a coordinate basis (\partial_\alpha), where for each \alpha the basis vector has components

    \[(\partial_\alpha)^\mu = \delta_\alpha^\mu\]

in these coordinates. We also have the coordinate dual basis (dx^\beta) where each dual vector or “1-form” has components

    \[(dx^\beta)_\mu = \delta^\beta_\mu.\]

Now while these bases are dual in the sense of bases:

    \[dx^\beta(\partial_\alpha) = \delta^\beta_\alpha\]

(by definition of dual basis), the individual vectors are not dual to the individual 1-forms in the sense of individual vectors. That is, for any given \alpha, we have \partial_\alpha and dx^\alpha are not dual in general.

Instead, recall indices are raised and lowered using the metric components (in a coordinate basis, that is). Possibly the result could be seen by inspection, but for clarity let’s write \mathbf e := \partial_\alpha for some chosen \alpha. This vector has components e^\nu = \delta^\nu_\alpha, hence the corresponding 1-form has components e_\mu = g_{\mu\nu}e^\nu = g_{\mu\nu}\delta^\nu_\alpha = g_{\mu\alpha}. By the meaning of components this says the 1-form is g_{\mu\alpha}dx^\mu. This is not dx^\alpha, in general! In “musical isomorphism” notation, the result is:

    \[(\partial_\alpha)^\flat = g_{\mu\alpha}dx^\mu\]


    \[(dx^\alpha)^\sharp = g^{\mu\alpha}\partial_\mu.\]

To show the result another way, recall the metric defines the dual to our vector \partial_\alpha to be \mathbf g(\partial_\alpha,\cdot). To examine this 1-form, feed it a vector (specifically, basis vectors \partial_\mu) and see how it acts on it:

    \[(\partial_\alpha)^\flat(\partial_\mu) := g(\partial_\alpha,\partial_\mu) = g_{\alpha\mu},\]

which says (\partial_\alpha)^\flat = g_{\alpha\mu}dx^\mu, as before.

In closing, another reason we cannot have (\partial_\alpha)^\flat = dx^\alpha in general is that the coordinate basis vector \partial_\alpha is not defined in terms of x^\alpha alone, but also all the other coordinates chosen. More on that next.

(Schutz (2009, §3.3, §3.5) makes a superb background to this discussion, and while the cited sections are for special relativity, in this case you can simply replace the Minkowski metric \boldsymbol\eta with an arbitrary curved metric \mathbf g.)

Mimicking a black hole in flat spacetime

For a Schwarzschild-Droste black hole, the curvature of 3-dimensional space is often depicted as a funnel shape (Flamm 1916 ). As I emphasise in forthcoming papers, this assumes the static slicing of spacetime, whereas other slicings yield different embedding diagrams. This leads to the question, could we slice flat spacetime in such a way that we get a similar funnel, or mimic other properties of a black hole? While this cannot of course change the fact the 4-dimensional spacetime is flat, the point is there is much flexibility in defining the 3-space, because it depends only on the chosen slicing or observers.

Embedding diagram for a fake black hole, representing an unusual spatial slice of Minkowski spacetime
Spoiler: Yes you can! This is an embedding diagram for our “fake black hole”, representing an unusual spatial slice of Minkowski spacetime. This looks more like a spinning top than a funnel.

Let’s start with Minkowski spacetime in spherical coordinates:

    \[ds^2 = -dt^2 + dr^2 + r^2(d\theta^2+\sin^2\theta\,d\phi^2)\]

This defines an inertial frame. Now suppose spacetime is filled with test particles moving radially, relative to the coordinate origin. Take coordinate speed dr/d\tau = \pm\sqrt{2M/r}, by analogy with the Schwarzschild and even Newtonian cases (choose one sign and stick with it). The 4-velocity is then:

    \[u^\mu = \bigg(\sqrt{1+\frac{2M}{r}},\pm\sqrt\frac{2M}{r},0,0\bigg)\]

which follows from normalisation \mathbf u\cdot\mathbf u=-1. Next we define a new time coordinate. A natural first attempt is to try the proper time of the particles. This may be obtained via local Lorentz boosts, or equivalently by a neat trick of lowering the index on the 4-velocity vector then taking its negative:

    \[-\mathbf u^\flat = \sqrt{1+\frac{2M}{r}}dt \mp\sqrt\frac{2M}{r}dr\]

(I explain this approach in a forthcoming paper, but it is inspired by Martel & Poisson 2001  and ultimately based on Frobenius’ theorem: see the variant for 1-forms described in de Felice & Clarke §2.12.) Expressing the dual velocity this way, as an explicit sum of the coordinate dual basis vectors dt and dr, is suggestive of a total differential which we would hope is the proper time d\tau. Unfortunately the expression is not a total differential, as seen by examining the coefficient of dt. But from inspection we can use an integrating factor: divide through by \sqrt{1+2M/r}, simplify, and define the resulting expression as the differential of a new time coordinate T:

    \[dT := dt \mp\frac{1}{\sqrt{1+r/2M}}dr\]

(Incidentally, this easily integrates to T = t \mp 4M\sqrt{1+r/2M} plus a constant of integration.) While T is not the proper time, its level sets T=\textrm{const} coincide with the 3-space of the observers as shown next, which is sufficient for our embedding diagram. Since \mathbf u is by definition orthogonal to its local 3-space, the dual vector \mathbf u^\flat is also normal to this 3-space. But dT is parallel to \mathbf u^\flat, hence they are normal to the same 3-space, but any gradient dT is normal to the level sets T=\textrm{const}, which proves the claim.

This is analogous to static observers in Schwarzschild spacetime. While the Schwarzschild t-coordinate is not their proper time, setting t=\textrm{const} still determines the same 3-space as these observers. Also we cannot replace the t-coordinate with proper time while still retaining the coordinates r, \theta, and \phi. The derivative dT/d\tau = 1/\sqrt{1+2M/r} for our fake black hole is also reminiscent of static observers in Schwarzschild spacetime.

Rearrange the earlier expression for dT and substitute into the line element to obtain:

    \[ds^2 = -dT^2 \mp \frac{2}{\sqrt{1+r/2M}}dT\,dr + \bigg(1+\frac{2M}{r}\bigg)^{-1}dr^2\]

plus the 2-sphere metric r^2(d\theta^2+\sin^2\theta\,d\phi^2). These coordinates have no issue at r=2M, and while there is a coordinate singularity at r=0 the metric was degenerate there even in our initial spherical coordinates. The Riemann tensor is zero, as it must be since this is still flat spacetime. Since g^{TT}=-(1+2M/r)^{-1} the coordinate T is timelike everywhere. The 4-velocity in the new coordinates is u^\mu=(1/\sqrt{1+2M/r},\pm\sqrt{2M/r},0,0). Integrating dT/dr = u^T/u^r gives the travel time T(r) which is well behaved unlike Schwarzschild t(r) which diverges. The radial proper distance for our test particle observers is (1+2M/r)^{-1/2}dr, which gets very small for r\ll 2M compared to the inertial frame which measures radial distance dr everywhere.

A typical isometric embedding diagram for a spherically symmetric spacetime takes a slice of constant “time”, here T=\textrm{const}, through the equator \theta=\pi/2. This is matched isometrically with a surface z=z(r) in a 3-dimensional flat space. The flat space is taken to be Euclidean or Minkowski space, with the metric dr^2+r^2d\phi^2\pm dz^2 in cylindrical coordinates (the sign is unrelated to our previous sign choice). Our case requires the minus sign for Minkowski space since g_{rr}<1. It follows z = 4M\sqrt{1+r/2M}, which may be plotted in a scale-invariant way as z/M against r/M.

Embedding diagram for a fake black hole, an unusual choice of spatial slice in Minkowski spacetime
The same embedding diagram from a lower viewpoint and with further comments. The diagram is the same for both ingoing and outgoing particles / observers. This surface extends to the origin, unlike Flamm’s paraboloid for Schwarzschild space which corresponds to static observers and hence is only defined outside r=2M.

The particles must be accelerating, as their motion is not caused by gravity. In the new coordinates ingoing particles have 4-acceleration (0,-M/r^2,0,0), outgoing particles have a different expression, but both have magnitude a=(1+2M/r)^{-1/2}M/r^2. Again these expressions are reminiscent of static observers in Schwarzschild spacetime. Each particle has a “Rindler” horizon at distance 1/a as measured in the instantaneous comoving frame, so in the original inertial frame this is contracted by the Lorentz factor \gamma=\sqrt{1+2M/r} and occurs at position r\mp r^2/M (simultaneous in the instantaneous comoving frame).

The kinematic decomposition of the particle worldlines yields zero vorticity, which is fortunate because by Frobenius’ theorem this is the condition for the local 3-spaces to all patch together consistently. The expansion tensor, expressed in the frame of the particles (different frames for ingoing and outgoing), is \mp\sqrt{M/2r^3} in the radial direction, and \pm\sqrt{2M/r^3} in the tangential directions. The shear is twice this amount in the radial direction, and half this amount in the tangential directions.

In the new coordinates the lapse is \sqrt{1+2M/r} and the shift (\mp 2M/r\cdot\sqrt{1+r/2M},0,0). The extrinsic curvature (of the 3D spatial slices inside 4D Minkowski spacetime, not the 2D embedded slice) is \pm\sqrt{2M/r^3} times -dr^2/2(1+2M/r)+r^2(d\theta^2+\sin^2\theta\,d\phi^2). This has trace K = \pm\sqrt{M/2r^3}(1/(1+2M/r)-4) or \pm\sqrt{M/2r^3}(3+8M/r)/(1+2M/r).

Finally, Flamm’s paraboloid is an iconic image, and I defend visualisations and metaphors in general as helpful and intuitive. But one should understand the limitations, in contrast to Painlevé 1921  for example who found a slicing of Schwarzschild spacetime into Euclidean 3-spaces \mathbb R^3, but drew some overly zealous conclusions from this (thanks to Andrew Hamilton for discussion on this point). Admittedly the static slicing in Schwarzschild spacetime is a natural choice, while my “fake black hole” slicing is contrived. But still, the reproduction of a funnel-shaped embedding in flat spacetime shows the need for caution in interpreting Flamm’s paraboloid as gravity.

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 (§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’ (§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 §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 , §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.

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.

black holes poster 2017

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 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”.