Suppose we have a coordinate system . This defines a coordinate basis , where for each the basis vector has components
in these coordinates. We also have the coordinate dual basis where each dual vector or “1-form” has components
Now while these bases are dual in the sense of bases:
(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 , we have and 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 for some chosen . This vector has components , hence the corresponding 1-form has components . By the meaning of components this says the 1-form is . This is not , in general! In “musical isomorphism” notation, the result is:
To show the result another way, recall the metric defines the dual to our vector to be . To examine this 1-form, feed it a vector (specifically, basis vectors ) and see how it acts on it:
which says , as before.
In closing, another reason we cannot have in general is that the coordinate basis vector is not defined in terms of 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 with an arbitrary curved metric .)
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.
Let’s start with Minkowski spacetime in spherical coordinates:
This defines an inertial frame. Now suppose spacetime is filled with test particles moving radially, relative to the coordinate origin. Take coordinate speed , by analogy with the Schwarzschild and even Newtonian cases (choose one sign and stick with it). The 4-velocity is then:
which follows from normalisation . 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:
(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 and , is suggestive of a total differential which we would hope is the proper time . Unfortunately the expression is not a total differential, as seen by examining the coefficient of . But from inspection we can use an integrating factor: divide through by , simplify, and define the resulting expression as the differential of a new time coordinate :
(Incidentally, this easily integrates to plus a constant of integration.) While is not the proper time, its level sets coincide with the 3-space of the observers as shown next, which is sufficient for our embedding diagram. Since is by definition orthogonal to its local 3-space, the dual vector is also normal to this 3-space. But is parallel to , hence they are normal to the same 3-space, but any gradient is normal to the level sets , which proves the claim.
This is analogous to static observers in Schwarzschild spacetime. While the Schwarzschild -coordinate is not their proper time, setting still determines the same 3-space as these observers. Also we cannot replace the -coordinate with proper time while still retaining the coordinates , , and . The derivative for our fake black hole is also reminiscent of static observers in Schwarzschild spacetime.
Rearrange the earlier expression for and substitute into the line element to obtain:
plus the 2-sphere metric . These coordinates have no issue at , and while there is a coordinate singularity at 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 the coordinate is timelike everywhere. The 4-velocity in the new coordinates is . Integrating gives the travel time which is well behaved unlike Schwarzschild which diverges. The radial proper distance for our test particle observers is , which gets very small for compared to the inertial frame which measures radial distance everywhere.
A typical isometric embedding diagram for a spherically symmetric spacetime takes a slice of constant “time”, here , through the equator . This is matched isometrically with a surface in a 3-dimensional flat space. The flat space is taken to be Euclidean or Minkowski space, with the metric in cylindrical coordinates (the sign is unrelated to our previous sign choice). Our case requires the minus sign for Minkowski space since . It follows , which may be plotted in a scale-invariant way as against .
The particles must be accelerating, as their motion is not caused by gravity. In the new coordinates ingoing particles have 4-acceleration , outgoing particles have a different expression, but both have magnitude . Again these expressions are reminiscent of static observers in Schwarzschild spacetime. Each particle has a “Rindler” horizon at distance as measured in the instantaneous comoving frame, so in the original inertial frame this is contracted by the Lorentz factor and occurs at position (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 in the radial direction, and 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 and the shift . The extrinsic curvature (of the 3D spatial slices inside 4D Minkowski spacetime, not the 2D embedded slice) is times . This has trace or .
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 , 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.
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 , and a frame with components in the coordinate system, that is:
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 for the vector . (In place of our , de Felice & Clarke write , and Carroll swaps the index order to .) In a CAS we can implement the frame as a array / matrix called “” say, reading the indices of from left to right but ignoring their up-or-down placement. This ordering conveniently gives an array of “vectors”:
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 implemented as a matrix ““, which give the coordinate basis vectors in terms of the new frame:
It follows from linear independence that , hence as matrices: . The transpose is required because of the index summation order, since the convention for matrix multiplication is . 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: .
Now suppose a vector is specified by its coordinate basis components , which we implement as a 4-element array . Since , substituting the previous expression for and using linear independence gives the components in the new frame (note the Latin index) as: . Programmatically this is the matrix multiplication , at least for my CAS does not distinguish between a row and column vector but automatically matches the dimensions. Now suppose we have different vectors, stored in an matrix say (typically ). These are processed in a batch operation by converting to column vectors, applying the transformation, then transposing back, so the components are: , in the new frame.
Now consider the dual bases. In the coordinate dual basis, the vector dual to has components . These components can be implemented as an array where is the matrix and the row / column vector distinction is ignored as before. Again we can lower multiple vectors in one step via .
The dual to the new frame satisfies by definition, hence
which may be validated by substitution, and these components are just again. Similarly
which are the components again. Carroll’s description for an orthonormal frame is true for any frame:
The vielbeins  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):
which I interpret as a definition. This evaluates to (or ), hence . Define also
which evaluates to (or , where is the matrix ), hence . 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
which can be implemented as . In the particular case of an orthonormal frame , so in this case Latin indices are raised and lowered with the Minkowski metric. The metric in the dual frame is
so define . These are matrix inverses: . We can show Latin indices are raised or lowered using this frame metric, so for example .
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: and , 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 made from the “frame” components is also just the identity (see Carroll), so it and the metric tensor are equal. Input and into this tensor and it indeed returns . 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)
My CAS multiplies higher rank “matrices” by contracting the last index of with the first index of . Hence we can only change two indices of 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.
Suppose a test particle undergoes circular motion in Schwarzschild spacetime, and not necessarily at the freefall rate. We assume its worldline is “centred on ” so to speak, or to be technical we could speak of the integral of a Killing vector field which is timelike at the given location: this also includes the case of no rotation at all but that’s fine. For simplicity, orient Schwarzschild-Droste coordinates so the motion coincides with the “equator” , and also with increasing -coordinate. Note the -coordinate is constant. Define the angular velocity as the coordinate ratio
assumed to be constant. We can solve for the 4-velocity of the particle:
which by definition is also the “time” vector of the particle’s orthonormal frame. We must have
for the motion to remain timelike. Note that while freefall circular motion requires to be outside the photon sphere, for accelerated circular motion any is permissible.
Now for the “space” vectors, it is natural, given our orientation of coordinates, to define two of them as simply normalised coordinate vectors:
This fixes the final vector in the tetrad up to orientation (overall factor), so we choose the orientation with positive -component:
Then as required, where is the Kronecker delta function. The particle’s 4-acceleration is given by the covariant derivative . I omit the result, but only the -component is nonzero, which is not unexpected. The length of the 4-acceleration vector is the magnitude of proper acceleration:
For the special case of geodesic motion the acceleration must vanish, so , and the factor reduces to . See Hartle ( §9.3) for a very different derivation of this. For the expression reduces to the familiar acceleration of a static observer.
It is often said Newton was fortunate to define force on a particle as the change in momentum , not from the change in velocity , because the former generalises better. Here the momentum is , and clearly the expressions for force coincide if the mass 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:
However this expression is only valid in a local inertial frame (LIF), as Hartle (2003 , §20.4) clearly qualifies. Recall, the 4-momentum is where is the 4-velocity of the particle. We can split the force into two orthogonal vectors:
where (LIF) is the 4-acceleration. The 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 . The term containing 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” 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 (LIF) generalises to the covariant derivative . Hence, the fully covariant expression for 4-force is:
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:
since is the usual fully covariant expression, and of a scalar is . This expression for force is the same as the specific LIF case above.
is a Killing tensor field for Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime, which models a homogeneous and isotropic universe. We can use it to conveniently derive the cosmic redshift of photons, or decay of “peculiar velocity” for particles with mass. is the metric tensor, is the “scale factor”, and is the 4-velocity of observers comoving with the Hubble flow with components in the usual coordinate choices.
is conserved along a geodesic worldline (more precisely, is the tangent vector under some affine parameter). Such a conserved quantity exists because Killing tensors correspond to symmetries of spacetime.
We can evaluate
A photon has norm-squared , and energy as measured by a Hubble observer (assuming you chose the affine parameter to give the 4-momentum). A massive particle has norm-squared , and Lorentz factor as measured by a Hubble observer (assuming the affine parameter is proper time). The usual results follow easily. In my view this the most elegant derivation.
Note you can tweak some quantities, because multiplying a conserved quantity by a constant gives another conserved quantity, and the same applies for a Killing vector/tensor. So we may use the unnormalised scale factor in place of , because is also a Killing tensor, with conserved quantity . As another example, for a timelike particle with constant mass , we may define which is also conserved.
Curiously, the tensor contains the spatial projector for the Hubble observers. This is just the “space” part of the metric, along the usual homogeneous and isotropic spatial slices. We can rearrange the conserved quantity to give: , then take the square root. In words, the length of the spatial part of the tangent vector is inversely proportional to the scale factor. More simply, Hubble observers measure things to lessen? (I need to think about the spatial geodesic case more…) These results sound familiar and unremarkable, but are rigorous and general.
The only textbook I know which identifies it as a Killing tensor is Carroll (2004, §8.5) . (However Wald (1984, §5.3a) deserves credit for using a Killing vector in the same fashion. See my forthcoming post on FLRW Killing vectors.) One source which gives the tensor is Maharaj & Maartens (1987a, §4) and (1987b, §3) . (While they identify the tensor, and the resulting conserved quantity which they interpret as linear momentum, the term “Killing tensor” is not mentioned.) Also there is similar content in much older sources, including Robertson & Noonan (1968) .
The coordinate choice should not matter, as we can easily write the expression in a tensorial way. One can check that indeed , where the parentheses denote symmetrisation of the indices.)
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.]
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” measure a radial distance . 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 .
- 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.
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”.
Last year, NASA/ESA released a giant image of the Andromeda galaxy taken by the Hubble Space Telescope. At 4.3 Gb and 1.5 billion pixels this composite of 411 images is completely impractical for most of us, but fortunately one random internet denizen created a stunning panning video:
The Andromeda galaxy is the nearest large galaxy to our own galaxy, the Milky Way. (There are also several dozen smaller galaxies in our “Local Group”). Even though it is 2 million light years away, you can see Andromeda with the naked eye. In the video, the scenery gets brighter towards the end, as the view approaches the galactic centre where there are more stars. NASA has more details, and you can also download the image in various sizes or use a zoomable browser tool.