A Tensors in null basis

[Antarres]

I'm having a somewhat silly confusion, about interpretation of components of tensors in null basis(such as Newman-Penrose or similar formalisms).

For example, let's say that we're working in a 4d spacetime, and we have a null real orthonormal tetrad, consisting of null vectors $k$ and $\ell$, and two spacelike vectors. The following relations(inner products) hold:
$$k \cdot k = 0 \qquad k \cdot \ell = 1 \qquad \ell \cdot \ell = 0$$

Now if we consider some tensor field on this manifold, let's say that it's a rank 2 covariant tensor field $T_{ab}$. Then I am confused by the interpretation of the contraction $T_{ab}k^a k^b$. I mean this in a sense that this contraction would usually be interpreted as a component of this tensor in the 'k-k' direction.
But for example, if I expand this tensor in the basis:
$$T_{ab} = A*k_a k_b + \dots$$
Then extracting this coefficient $A$, we have $A = T_{ab} \ell^a \ell^b$. But isn't this coefficient $A$ actually the component in 'k-k' direction? I'm pretty sure that my logic is mistaken somewhere.

In non-null case, the first interpretation would always be true, it seems natural, however, for some reason it confuses me since one could say that 'canonically' $k_a$ is dual to $\ell^a$, while metrically, $k_a$ is obviously dual to $k^a$.

It's important to rectify this, since in cases where these null vectors are used(for example in the analysis of null hypersurfaces), they usually have quite different physical meaning.
So which of these two interpretations is true?

 

[Orodruin]

Tensor components are not extracted by contraction with the basis they are written in, they are extracted by contraction with the dual basis.

 

[cianfa72]

Tensor components are not extracted by contraction with the basis they are written in, they are extracted by contraction with the dual basis.

For the tensor's slots of "co-vector" type, you must fill them with basis elements of the bi-dual $V^{**}$. However, since $V$ and $V^{**}$ are canonically identified, you can fill those slots with elements of the chosen basis in $V$.

 

[Antarres]

My question wasn't about how to extract the right component. Of course this is done using a dual basis. It was more about the interpretation. Like for example, if I contract with a null vector $k$ that's part of a null orthonormal frame, then this component would be a component along 'transverse direction', since null vectors are mixed in the metric.
Meanwhile, if I have a usual orthonormal frame, and I contract by a basis vector(with a lowered index, so a metric dual), I will get the component that is along this same basis vector. So I was wondering if, conventionally, I would call the contraction with $k$ as a component 'along $\ell$', or would I call it as the component 'along $k$'. I guess that the first answer is more appropriate, but sometimes I saw it interpreted according to the second option, so I wondered what is the right description.

 

[PAllen]

My question wasn't about how to extract the right component. Of course this is done using a dual basis. It was more about the interpretation. Like for example, if I contract with a null vector $k$ that's part of a null orthonormal frame, then this component would be a component along 'transverse direction', since null vectors are mixed in the metric.

So what is a null orthonormal frame? I don’t see how, in Minkowski space, to have a set of 4 mutually orthogonal vectors, when one or more of them are null.

 

[PeterDonis]

a null orthonormal frame

There is no such thing. The two null vectors will not be orthogonal to each other, nor will they be normalized (since null vectors can't be normalized). Null frames simply don't have the same intuitive physical interpretation that standard orthonormal frames with one timelike and three spacelike basis vectors do.

 

[pervect]

Pick your favorite textbook (I like MTW's text Gravitation) and follow along for the textbook's conventions (which are usually pretty standard, though they can vary a bit) about how to say things. I don't think I've ever seen a formal textbook talk about a component "along a direction" - so I have to guess a bit as to what this might mean. My basic recommendation is that if you want to be clear and formal, don't use this language of "along a direction".

Unfortunately - or fortuantely? - my longer post ran into formatting difficulties with PF's latex, so I'll give a short summary.

You can simplify your thinking and clarify your question by talking about the inner product rather than using basis specific language. If a and b are vectors, then the inner product $a \cdot b$. You can informally say that this inner product is the component of b in the direction of a, or vica-versa I suppose - I think that's your question? But if you want to be formal, I would just talk about the inner product. You can certainly write the inner product of two vectors in terms of their components in some particular basis, but the concept of "inner product" exists independent of the basis choice and hence it is arguably simpler to just talk about the geometric, coordinate-independent idea of the inner product.

As you've already pointed out in your first post, your choice of basis vectors isn't orthogonal or normal. So just call it a basis and leave it at that. To me this is an example of over-writing, I take Bohr's dictum "Never express yourself more clearly than you are able to think" as an invitation to write less, rather than more.

 

[Antarres]

I apologize for the long pause in replies, I was kept away from the computer recently.

So to answer @PAllen and @PeterDonis, terminology 'null orthonormal frame' can surely be called an abuse of standard terminology. However, I've seen it used in papers and textbooks where this formalism is used. Since I never saw this formalism as part of a standard course in GR(but rather maybe some advanced topics), there's no point in referring to all the places where it might be found, it is just jargon.

The last place where I've read it would be https://arxiv.org/abs/1306.2517 page 12.

So to clarify, no, it is of course not possible to have four null vectors as a basis in Minkowski space, provided the vectors are real. If the vectors can be taken to be complex, then a basis can be formed of two real and two complex null vectors. This is usually called a complex null tetrad, or Newman-Penrose(NP) tetrad.

If one avoids taking the complex vectors, one can construct a basis of two null vectors and two spacelike vectors, with defining relations given in the OP. This is then called 'null orthonormal tetrad' often, or 'real null tetrad' so that it is different from NP tetrad.

I'm aware that null vectors don't have the same intuitive interpretations as standard timelike/spacelike ones, and the normalization here is taken to mean the relations defined in the OP rather than standard way of normalizing which is impossible for null vectors.

@pervect Your advice is maybe a good way to approach this. 'Along a direction' is a word that I would personally use to mean something that means 'direction'. In literature concerning hypersurfaces, often normal and tangential is used, or transverse in case we have null hypersurface. I was just wanting to make sure that a certain component(which is sometimes exchanged for a scalar product to make it manifestly invariant) can be said to point 'away from hypersurface' or such, as a physical interpretation. Maybe just not giving it a name is a safe choice, but yes, all I was focusing on was about interpreting certain components, it is quite clear how to calculate them, so that was not something that I focused on in the question.

 

[pervect]

When talking about directions, I've had arguments in the past about whether a "spacelike" direction is a vector, like $\partial / \partial x$, or a dual vector, like dx. This doesn't make much difference in a true orthonormal basis, but it does in other cases and can cause arguments.

The insight I have about null geodesics is that they have an affine parameterization, which means that while any interval along a null geodesic has a length of zero, it is still possible via means of the affine parameterization to mark off "equal intervals" along the geodesic. There's even a geometrical construction technique I'm rather fond of called "Schild's ladder", that describes in detail how to draw geometric diagrams to perform the marking procedure along any geodesic which is applicable to null geodesics. The full technique of Schild's ladder goes beyond this, describing not only how to do regular markings but how to do parallel transport (for a Levi-Civita connection), but the technique incorporates a principle of how to "mark off" equal intervals.

These intervals along a null geodesic are not time intervals, nor are they space intervals, so they're not really intuitive and I haven't found any way to make them really intuitive in terms of any non-mathematical description. I rely on the math for my intuition here.

It's not intuitive that, given the length of any segment of a null geodesic is equal to zero, that there is some sense in which segments can be said to have "the same" length and others to have "different lengths". But an affine geometry gives the necessary mathematical structure to perform this task, and null geodesics do have an associated affine geometry that can perform this task.

People who try to find the "viewpoint of a photon" typically make the mistake of trying to think of a null geodesic with markings of regular intervals (events at regular intervals) along as having some "duration" and/or "length" between the markings/events. But the math says something different.

I also like the idea of using two laser beams that interfere constructively and destructively to go from the idea of null coordinates to orthonormal coordinates.

The frequency of a laser beam is not a property of the beam alone, but a property of the beam and the choice of reference frame. Given two different laser beams that we postulate as geometric entites, one can find a frame of reference in which the frequencies of both beams are equal. If one adds in the necessary phase information as part of the description of the laser beam, these beams will create nodes of destructive (and constructive) interference which create a latice structure of rulers and clocks (for instance, the distance between the destructive nodes acts as a ruler, the period of oscillation of a constructive interference node serves as a clock).

While I find this idea personally helpful, it's obviously not very formal, and thus it's unclear if it will help anyone else. Specifically, when I talk about "laser beams", a picture springs to my mind that makes sense to me, but I don't have a formal mathematical description of this picture. I can say that these laser beams are supposed to be geometric entities, and that given a specific frame of reference, the "laser beams" must have both a frequency and a phase at any point in the frame of reference.

In spite of the lack of a more formal description, I rather like it - and I'll present it on the chance that some reader will find it useful.

 

[PeterDonis]

I've had arguments in the past about whether a "spacelike" direction is a vector, like $\partial / \partial x$, or a dual vector, like dx.

In what context? A "direction", whether it's spacelike, timelike, or null, corresponds to a vector (or more precisely a tangent vector, once we are dealing with curved manifolds and tangent spaces), and a vector corresponds to a directional derivative. MTW goes into this in detail. In the absence of a metric, which lets you lower indexes, there is no natural correspondence that I'm aware of between directions and dual vectors (1-forms as MTW calls them).

The insight I have about null geodesics is that they have an affine parameterization, which means that while any interval along a null geodesic has a length of zero, it is still possible via means of the affine parameterization to mark off "equal intervals" along the geodesic.

Sure, this is basic affine geometry. Affine parameterization does not have to correspond to "length" in terms of a metric.

There's even a geometrical construction technique I'm rather fond of called "Schild's ladder", that describes in detail how to draw geometric diagrams to perform the marking procedure along any geodesic which is applicable to null geodesics. The full technique of Schild's ladder goes beyond this, describing not only how to do regular markings but how to do parallel transport (for a Levi-Civita connection), but the technique incorporates a principle of how to "mark off" equal intervals.

Yes, I wish more GR textbooks went into this in detail, as MTW does. I'm surprised Wald doesn't.

These intervals along a null geodesic are not time intervals, nor are they space intervals, so they're not really intuitive and I haven't found any way to make them really intuitive in terms of any non-mathematical description.

The actual affine lengths don't have any natural physical interpretation, because there is no natural affine parameterization that's picked out by the physics, the way there is for timelike and spacelike curves. If you introduce timelike inertial observers and start talking about the frequencies of radiation they observe, then you can correlate this with affine parameterizations of null geodesics that arise from the natural inertial coordinate charts associated with the observers. But the null geodesics by themselves don't give you any such structure.

The frequency of a laser beam is not a property of the beam alone, but a property of the beam and the choice of reference frame.

I would say the beam and the observer who is either emitting or receiving the beam.