Black Holes - the two points of view.

[TrickyDicky]

Not for the standard definition of "static", which is that the spacetime (or a region of it) has a timelike KVF. That definition is coordinate-independent. By that definition, Schwarzschild spacetime (the vacuum solution) is static only outside the EH; the region at and inside the EH is not static.

That standard definition also says that the timelike KVF is global for the spacetime. Please explain what you think global means in this context. And if you consider the maximally extended Schwarzschild spacetime to be (globally) static or non-static.

There is also a coordinate-independent definition of "expansion", but it applies to families of timelike curves, not "space" itself. The standard definition of an "expanding" or "contracting" FRW spacetime uses the family of timelike curves that describe the worldlines of "comoving" observers; the expansion of that family of curves, defined in the standard coordinate-independent way, is positive for an expanding FRW spacetime and negative for a contracting one.

You are confirming here that there is no frame-independent (here frame is used in both its meaning of coordinate system and observer state of motion senses) definition of non-static spacetime since it relies in a family of comoving observers, so I don't know in what sense you call it coordinate-independent.

Not by the standard definition of "homogeneity", the one that applies to FRW spacetimes. Can you give a reference for this different definition of "homogeneity"? The standard term for what you're describing, AFAIK, is simply "spherical symmetry".

Give me your standard definition of homogeneity or stop referring to it. Spherical symmetry only requires a foliation of concentric 2-spheres around an origin. A foliation in which each point is 2-sphere has an origin for each point. Can't you see that?

The "volume" inside the EH is not necessarily finite; as I said before, it depends on what "volume" you are looking at. The 4-volume inside the EH is infinite, since it covers an infinite range of the t coordinate. Spacelike 3-volumes cut out of that 4-volume may be finite or infinite, depending on how they are cut.

We were talking clearly about the 3-space volume. If observers arrive at the singularity in a finite time i guess for them the volume is finite.

The portion of the spacetime that is occupied by the matter that originally collapsed to form the BH *is* full of matter on its way to the singularity. And this portion (at least in the idealized spherically symmetric case) *is* isometric to a portion of a collapsing FRW spacetime. That is the model that Oppenheimer and Snyder described in their 1939 paper. However, that only applies to the non-vacuum portion of the spacetime. I don't really think an analogy between the *vacuum* portion of the spacetime inside the EH and FRW spacetime is useful, but that may be just me.

It's the first time you say that the Schwarzschild spacetime has a non-vacuum portion, are you sure? And how you separate the non-vacuum part of the BH from the vacuum part. I thought the whole spacetime was supposed to be a vacuum solution.

Staticity, by the standard definition, *is* coordinate-independent. See above.

Are you talking about static patches within a nonstatic spacetime or to spacetimes globally defines as static. What would you consider to be the case with Schwarzschild spacetime?

The KVF \partial / \partial t in Schwarzschild spacetime is not a "different KVF" in different regions. But any vector field on a manifold is a mapping between points in the manifold and vectors in a vector space, and different points may map to different vectors.
No. Whether a KVF, or indeed *any* vector field, is timelike, spacelike, or null *at a given event* is an invariant, independent of coordinates. But the particular vectors which are mapped to different events by a vector field are different vectors, and may have a different causal nature.

Ok, but have you tried to compute the KVFs of the de Sitter spacetime using first the static coordinates and then the nonstatic ones including the dS slicing?

 

[TrickyDicky]

I forgot to comment on this specifically. AFAIK, the standard definition of "homogeneity" for a spacetime is that there are 3 KVFs corresponding to spatial translations. The fact that those are *not* present in Schwarzschild spacetime is why I have been objecting to describing that spacetime as "homogeneous".

I entertained this idea as well. Do you know of any reference (or some heuristic explanation)that confirms that homogeneity requires 3 translation spacelike KVFs?

 

[erasrot]

Ok, let me be sure I understand this properly. Timelike geodesics are timelike everywhere in Schwarzschild spacetime (in the manifold, i.e., independent of the given chart), including the region inside the EH. This spacetime is geodesically incomplete, what implies that the affine parameter for any infalling observer is bounded from above so, after proper calculation, the difference between the bound for a given geodesic and any other value of the affine parameter along any other timelike geodesic is also finite. I am right?

 

[TrickyDicky]

I wouldn't say that. If your GR manifold is everywhere vacuum then there should not be preferred slicing, but if your GR manifold has matter fields then the matter fields will disrupt the symmetry and may define a preferred slicing.

Sure, but that would just give a special status to a certain frame, it would not break the general covariance of GR that demands coordinate -independence.

The V in KVF is "vector". The timelike or spacelike character of a vector has nothing to do with the coordinates chosen, they remain timelike or spacelike in any coordinate chart. If it were not so, then they would not be vectors.

Look up the static and non-static coordinates for de Sitter spacetime. Why would staticity (a property that includes admission of timelike Killing vector field) depend on the coordinates used?
Something similar happens with Schwarzschild vs Lemaitre coordinates for the region inside the EH.

 

[PeterDonis]

I entertained this idea as well. Do you know of any reference (or some heuristic explanation)that confirms that homogeneity requires 3 translation spacelike KVFs?

I don't have a reference handy right now but I'll check my copy of MTW as soon as I'm able. I'm pretty sure they go into it when they discuss FRW spacetime. In any case, the fact that FRW spacetime has 3 spacelike translation KVFs which Schwarzschild spacetime does not is certainly a fact.

That standard definition also says that the timelike KVF is global for the spacetime. Please explain what you think global means in this context. And if you consider the maximally extended Schwarzschild spacetime to be (globally) static or non-static.

Why do we keep having to harp on terminology instead of physics? I've already given the physics, several times, and so have others: the 4th KVF on Schwarzschild spacetime (the one in addition to the 3 that arise from spherical symmetry) is timelike outside the horizon, null on the horizon, and spacelike inside it. I haven't used the word "global", and whether that word appears in the standard definition depends on whose definition you read. I don't think MTW use the word (but I'll check when I can). This, once again, is why I have said it's no good just reading the words sources use about these things; you have to look at the actual math, which I have given.

If you want a guess as to why the word "global" appears, it's because the sources you're reading don't draw a clear distinction between an entire, maximally extended manifold, and a the region of that manifold that is covered by a particular coordinate chart without coordinate singularities. So when they say the 4th KVF is "globally" static, they really mean "static over the entire region covered by the exterior Schwarzschild chart", which is just the region outside the horizon. Someone who didn't realize that the exterior Schwarzschild chart doesn't cover the entire maximally extended manifold (and there have sure been plenty of them posting on PF) might think that static region was the entire manifold; and someone who didn't stop to think about that possible misinterpretation might use the word "global" in the sloppy sense I have described. But that's just a guess; I don't know what the people who used the word "global" were thinking.

You are confirming here that there is no frame-independent (here frame is used in both its meaning of coordinate system and observer state of motion senses) definition of non-static spacetime since it relies in a family of comoving observers, so I don't know in what sense you call it coordinate-independent.

The definition of "expansion" I gave applies to a family of timelike worldlines, but the definition of "static" that I gave does *not*; it applies to a region of spacetime, not a family of curves in that region, and whether or not a given region of spacetime is or is not static is a coordinate-independent fact.

It is perfectly possible to have a region of spacetime which is static but has families of timelike curves in it that have nonzero expansion, so there is no necessary connection between a region being static and the expansion of families of timelike curves within that region. If you insist on using the word "static" in a non-standard way, as meaning "zero expansion", I suppose I can't stop you, but please don't read *me* as using it that way.

(Also, once a family of timelike curves is defined, its expansion is coordinate-independent; it comes out the same regardless of which chart you use to describe the curves. But that's a secondary point.)

We were talking clearly about the 3-space volume.

Yes, but *which* 3-space?

If observers arrive at the singularity in a finite time i guess for them the volume is finite.

It depends on which 3-space volumes you pick. If you pick the 3-space volumes defined by constant r, theta, phi and the full range of Schwarzschild time t, then each such 3-space volume is infinite; and an infalling observer passes through a range of such 3-volumes between the horizon and the singularity (each one labeled by a different r--we're assuming a radially infalling observer). If you pick the 3-space volumes defined by constant Painleve time T, constant theta, phi, and 0 < r < 2m, then each such 3-space volume is finite, and an infalling observer also passes through a range of these 3-volumes (each one labeled by a different T) between the horizon and the singularity.

It's the first time you say that the Schwarzschild spacetime has a non-vacuum portion, are you sure?

I was talking there about the Oppenheimer-Snyder model, which has a non-vacuum portion representing spherically symmetric collapsing dust, joined to a portion of Regions I and II of the maximally extended vacuum Schwarzschild spacetime, representing the vacuum region outside the surface of the collapsing matter. I realize that's a switch of model, but I only mentioned it because you brought up matter that originally collapsed to form the BH. If you include such matter at all, then you aren't talking any more about the full maximally extended vacuum Schwarzschild spacetime, but only the portions that I just described. Everything I've said about vacuum Schwarzschild spacetime still applies to those portions of Regions I and II that appear in the Oppenheimer-Snyder model.

Are you talking about static patches within a nonstatic spacetime or to spacetimes globally defines as static.

I'm talking about static spacetime regions. See above.

Ok, but have you tried to compute the KVFs of the de Sitter spacetime using first the static coordinates and then the nonstatic ones including the dS slicing?

No, but you're right, it's a good exercise. I'll take a look at it when I get a chance.

 

[PeterDonis]

Look up the static and non-static coordinates for de Sitter spacetime. Why would staticity (a property that includes admission of timelike Killing vector field) depend on the coordinates used?

Static coordinates are just coordinates that make the timelike KVF (in the regions where it is timelike) manifest, by explicitly having one coordinate that the metric is independent of. In the case of Schwarzschild spacetime described by the Schwarzschild chart, that coordinate is t. So any region of spacetime that is static can be described by a static coordinate chart.

However, the converse is not true; there is no reason why a static region *has* to be described by a static chart. Sometimes there are good reasons not to, as with Lemaitre coordinates on Schwarzschild spacetime. However, even in a non-static chart on a static spacetime region, if you describe the timelike KVF in the non-static chart, it will still satisfy Killing's equation--i.e., it will still be a KVF. It just won't be as obvious.

(I'll defer further remarks on this until I've had a chance to explicitly do the computation for de Sitter spacetime.)

Edit: I should probably add that even in regions where a given KVF is *not* timelike, there will still be analogues of the "static" chart--i.e., a chart in which the metric is independent of some coordinate corresponding to the KVF--though of course "static" is not a good name for the chart in those regions. An example is the Schwarzschild chart on the interior vacuum region of Schwarzschild spacetime: the metric is still independent of t, but t is not timelike and \partial / \partial t, while still a KVF, is not a timelike KVF.

 

[TrickyDicky]

If you want a guess as to why the word "global" appears, it's because the sources you're reading don't draw a clear distinction between an entire, maximally extended manifold, and a the region of that manifold that is covered by a particular coordinate chart without coordinate singularities. So when they say the 4th KVF is "globally" static, they really mean "static over the entire region covered by the exterior Schwarzschild chart", which is just the region outside the horizon. Someone who didn't realize that the exterior Schwarzschild chart doesn't cover the entire maximally extended manifold (and there have sure been plenty of them posting on PF) might think that static region was the entire manifold; and someone who didn't stop to think about that possible misinterpretation might use the word "global" in the sloppy sense I have described. But that's just a guess; I don't know what the people who used the word "global" were thinking.

I don't think so, my guess is that the language of the definitions comes from Riemannian geometry and there you don't have to make causal distinctions for vectors. When those definitions are applied to pseudoriemannian spacetimes certain conceptual problems appear due to the different causal vectors. And globality is one of those concepts that suffer with the introduction of these distinctions, the other I would say is the coordinate independence, not of the KVF itself, but of its causal nature.

It is perfectly possible to have a region of spacetime which is static but has families of timelike curves in it that have nonzero expansion, so there is no necessary connection between a region being static and the expansion of families of timelike curves within that region. If you insist on using the word "static" in a non-standard way, as meaning "zero expansion", I suppose I can't stop you, but please don't read *me* as using it that way.

Fine, but then you are implying that staticity and non-staticity can coexist in the same region, that are not mutually excluding concepts, right?

 

[Dale]

Sure, but that would just give a special status to a certain frame, it would not break the general covariance of GR that demands coordinate -independence.

Yes, exactly.

Look up the static and non-static coordinates for de Sitter spacetime. Why would staticity (a property that includes admission of timelike Killing vector field) depend on the coordinates used?

Whether or not the spacetime is static doesn't depend in any way on the coordinates. Static coordinates are a very different thing from a static spacetime. If you have a static spacetime then there exists a set of coordinates where the components of the metric are not functions of time, but you can always choose different coordinates where some of the components are. For example, Minkowski spacetime is obviously static, but you can use non-static rotating coordinates if you want. Doing so doesn't change any of the KVF's, but of course the components of the KVF's are not as easy to figure out in those coordinates.

 

[PeterDonis]

I don't think so, my guess is that the language of the definitions comes from Riemannian geometry and there you don't have to make causal distinctions for vectors. When those definitions are applied to pseudoriemannian spacetimes certain conceptual problems appear due to the different causal vectors.

This certainly could be the case, yes.

And globality is one of those concepts that suffer with the introduction of these distinctions

If you think "globality" is an important concept, yes, I suppose it could. I've never really thought of "globality" as a concept of interest.

the other I would say is the coordinate independence, not of the KVF itself, but of its causal nature.

I've already addressed this, and so have others. The causal nature of any vector at a particular event in any spacetime is coordinate-independent. The causal nature of vectors at *different* events, but which are part of the same vector field, can be different, but whether or not they are, and if so, how, is also coordinate-independent, once the particular events in question are specified.

Fine, but then you are implying that staticity and non-staticity can coexist in the same region, that are not mutually excluding concepts, right?

I'm not sure what you mean by this, but I have already described the physics many times, and I have explicitly given you the definition I am using for the word "static", and explicitly told you how it applies to Schwarzschild spacetime. Whatever question you are asking here, you should be able to answer it for yourself from what I have already written.

 

[TrickyDicky]

de Sitter space metric in static coordinates:

ds^2=-(1-\frac{r^2}{\alpha^2})dt^2+(1-\frac{r^2}{\alpha^2})^{-1} dr^2+r^2d\Omega^2_{n-2}

Schwarzschild spacetime in Lemaitre coordinates:
ds^2=d\tau^2-\frac{2\mu}{r}dρ^2-r^2(d\theta^2+sin^2\theta d\phi^2)

In both cases none of the metric coefficients are a function of time.

 

[Dale]

Schwarzschild spacetime in Lemaitre coordinates:
ds^2=d\tau^2-\frac{2\mu}{r}dρ^2-r^2(d\theta^2+sin^2\theta d\phi^2)

In both cases none of the metric coefficients are a function of time.

For Lemaitre coordinates, if by "time" you mean τ then the metric coefficients of ρ θ and phi are all functions of time since r is a function of time.

 

[PAllen]

de Sitter space metric in static coordinates:

ds^2=-(1-\frac{r^2}{\alpha^2})dt^2+(1-\frac{r^2}{\alpha^2})^{-1} dr^2+r^2d\Omega^2_{n-2}

Schwarzschild spacetime in Lemaitre coordinates:
ds^2=d\tau^2-\frac{2\mu}{r}dρ^2-r^2(d\theta^2+sin^2\theta d\phi^2)

In both cases none of the metric coefficients are a function of time.

except that r=(3/2(ρ - τ ))^2/3 / (2M)^1/3

So, all non-vanishing components of the metric are time dependent except g00.

 

[PAllen]

As for the static de Sitter coordinates, for which metric given in recent post, the key point is that static coordinates only cover a quarter of the spacetime. Thus the issue is similar to complete SC geometry. The portion of de-Sitter covered by the static patch really is static per timelike KVF definition, but the rest of the spacetime is not static.

 

[PeterDonis]

As for the static de Sitter coordinates, for which metric given in recent post, the key point is that static coordinates only cover a quarter of the spacetime.

Just to clarify, I believe that the line element TrickyDicky wrote down is valid over the entire spacetime, but the t coordinate, which is the one the line element is independent of (i.e., \partial / \partial t is the KVF) is only timelike in the region you mention, that covers a quarter of the spacetime. The rest of the spacetime is beyond the "cosmological horizon" and the t coordinate is spacelike there (and null on the horizon itself).

 

[PAllen]

Just to clarify, I believe that the line element TrickyDicky wrote down is valid over the entire spacetime, but the t coordinate, which is the one the line element is independent of (i.e., \partial / \partial t is the KVF) is only timelike in the region you mention, that covers a quarter of the spacetime. The rest of the spacetime is beyond the "cosmological horizon" and the t coordinate is spacelike there (and null on the horizon itself).

Ok, that makes sense, like the SC coordinates. I've only read things where they use different coordinates in the non-static regions, using the 'static' coordinates only in truly static section.

 

[PeterDonis]

Ok, that makes sense, like the SC coordinates. I've only read things where they use different coordinates in the non-static regions, using the 'static' coordinates only in truly static section.

Yes, but when comparing with SC coordinates, a key thing to remember is that there are an infinite number of "static" coordinate charts on de Sitter spacetime, because the line element that TrickyDicky wrote down can be centered on *any* spatial point (i.e., any spatial point can be designated as r = 0). Of course in Schwarzschild spacetime there is only one KVF which is timelike on any portion of the spacetime, and the only charts I'm aware of whose "time" coordinate matches that KVF are the Schwarzschild and Painleve charts.

As the above statement indicates, there are also an infinite number of timelike KVFs on de Sitter spacetime, just as there are on Minkowski spacetime; but in de Sitter spacetime, each timelike KVF has its own "static region" in which it is timelike, and which is bounded by its own cosmological horizon on which the KVF becomes null (and then spacelike beyond the horizon).

Actually, it may be even more complicated than that, because de Sitter spacetime has the same isometry group as Minkowski spacetime, and Minkowski spacetime actually has *two* infinite families of timelike KVFs. The first corresponds to the worldlines of inertial observers (at rest in all the different possible global inertial frames), and the second corresponds to the worldlines of Rindler observers (with all the different possible Rindler horizons, based on which event in the spacetime is chosen as the "pivot point" where the past and future horizons intersect). I will have to do some computations to figure out which of those two types of observers in Minkowski spacetime corresponds to the observers who are at rest in the static chart that TrickyDicky wrote down (in the region where it is possible to have static observers); I suspect the closest analogue is actually Rindler observers, because it looks to me like any observer who remains at a constant r in the static de Sitter chart (where 0 < r < alpha) will have nonzero proper acceleration.

 

[TrickyDicky]

except that r=(3/2(ρ - τ ))^2/3 / (2M)^1/3

So, all non-vanishing components of the metric are time dependent except g00.

For Lemaitre coordinates, if by "time" you mean τ then the metric coefficients of ρ θ and phi are all functions of time since r is a function of time.

Sure, I was looking only at the explicit dependence.

Ok, so substitute Lemaitre's with the Gullstrand–Painlevé coordinates as PeterDonis suggests.

 

[Dale]

OK, and what is the point? Are you trying to claim that these different coordinate charts change the KVFs somehow? If so, simply citing them is insufficient.

 

[TrickyDicky]

Just to clarify, I believe that the line element TrickyDicky wrote down is valid over the entire spacetime, but the t coordinate, which is the one the line element is independent of (i.e., \partial / \partial t is the KVF) is only timelike in the region you mention, that covers a quarter of the spacetime. The rest of the spacetime is beyond the "cosmological horizon" and the t coordinate is spacelike there (and null on the horizon itself).

My point was not to show that the t coordinate doesn't blow up a the cosmological horizon in the static coordinates, it does because they only cover a double wedge of the hyper-hyperboloid that represents the whole space.
My point was to show that this very region that is covered by the static coordinates and that admits a timelike KVF with this coordinates, when expressed in different coordinates (like the closed slicing that covers the entire space) doesn't admit the timelike KVF.

 

[Dale]

My point was to show that this very region that is covered by the static coordinates and that admits a timelike KVF with this coordinates, when expressed in different coordinates (like the closed slicing that covers the entire space) doesn't admit the timelike KVF.

You certainly didn't show that. To do that, you would have to write down the vector field and the metric in both coordinates, compute the Lie derivative of the metric wrt the vector field in each, and show that it is 0 in one and non-zero in the other.

 

[TrickyDicky]

You certainly didn't show that. To do that, you would have to write down the vector field and the metric in both coordinates, compute the Lie derivative of the metric wrt the vector field in each, and show that it is 0 in one and non-zero in the other.

I'm afraid you are not understanding what I'm saying, I have shown exactly what I said, not what you want me to show.
My point about coordinate-dependance was (as explained in a previous post) about the causal nature (the type if you will ) of the KVF, not about the existence or not of a KVF regardless of its causal character (timelike or spacelike).

 

[Dale]

My point about coordinate-dependance was (as explained in a previous post) about the causal nature (the type if you will ) of the KVF, not about the existence or not of a KVF regardless of its causal character (timelike or spacelike).

You didn't show that either. To do that, you would have to write down the vector field and the metric in both coordinates, compute the norm of the vector in each, and show that it has a different sign in one set of coordinates than in the other.

 

[TrickyDicky]

OK, and what is the point? Are you trying to claim that these different coordinate charts change the KVFs somehow? If so, simply citing them is insufficient.

No, as I said I'm only referring to the dependence of the causal character of a certain KVF on the coordinate chart.

 

[TrickyDicky]

You didn't show that either. To do that, you would have to write down the vector field and the metric in both coordinates, compute the norm of the vector in each, and show that it has a different sign in one set of coordinates than in the other.

For what I'm saying I think it is enough showing the dependence or lack of, of the metric coefficients of the line element (for coordinates covering the same region of the spacetime) on time.

 

[Dale]

Not if you are making a claim about the causal nature of the KVFs. Then you need to calculate the norm of the KVF in each coordinate system.

 

[TrickyDicky]

Maybe this reasoning helps to understand what I mean, of course if there is some flaw in it I'm sure you guys will point it out to me: different slicings of a spacetime ( in the case the spacetime is curved) cut the light cones in different ways altering the distribution of the inside and the outside of the cone and therefore of the vector field norms.
In the case the spacetime is flat like Minkowski's one has to chose patches from different regions of the spacetime to allow for non-staticity, see for instance the Milne universe that is a patch that corresponds to the interior of the Minkowski light cone and it is expanding.

But in the presence of curvature, one could (maybe not in all cases) pick the slicings that allow the same region of a spacetime to be static with one slicing and non-static with another.

 

[PAllen]

Maybe this reasoning helps to understand what I mean, of course if there is some flaw in it I'm sure you guys will point it out to me: different slicings of a spacetime ( in the case the spacetime is curved) cut the light cones in different ways altering the distribution of the inside and the outside of the cone and therefore of the vector field norms.
In the case the spacetime is flat like Minkowski's one has to chose patches from different regions of the spacetime to allow for non-staticity, see for instance the Milne universe that is a patch that corresponds to the interior of the Minkowski light cone and it is expanding.

But in the presence of curvature, one could (maybe not in all cases) pick the slicings that allow the same region of a spacetime to be static with one slicing and non-static with another.

No, I don't think any of this is true. A KVF is defined over spacetime. Different slicings and coordinates will relabel points, and modify the component expression of the KV at each point, but the timelike/spacelike nature of the KV at a particular event will never change.

 

[PAllen]

As the above statement indicates, there are also an infinite number of timelike KVFs on de Sitter spacetime, just as there are on Minkowski spacetime; but in de Sitter spacetime, each timelike KVF has its own "static region" in which it is timelike, and which is bounded by its own cosmological horizon on which the KVF becomes null (and then spacelike beyond the horizon).

Ok, now I am a bit confused. Both by definition (there exists a 'global' timelike KVF) and understanding (a timelike direction in which the metric doesn't change), any region that can be covered by these coordinates inside a cosmological horizon is static (since the KVF is also irrotational) , no scare quotes. The fact that some other coordinates don't manifest this in the metric expression shouldn't change this any more than Lemaitre coordinates cancel the static character of the exterior SC region.

 

[Dale]

different slicings of a spacetime ( in the case the spacetime is curved) cut the light cones in different ways

Yes, even in the case of a flat spacetime.

altering the distribution of the inside and the outside of the cone

No, the same events which are inside the cone remain inside the cone, as do the events which are outside. What changes is only which pairs of events are considered simultaneous or not.

and therefore of the vector field norms.

No, the norm of any given vector field at any given event will not change at all under any coordinate transform.

In the case the spacetime is flat like Minkowski's one has to chose patches from different regions of the spacetime to allow for non-staticity

Minkowski spacetime is static everywhere.

But in the presence of curvature, one could (maybe not in all cases) pick the slicings that allow the same region of a spacetime to be static with one slicing and non-static with another.

No, this is not the case. If there is a timelike KVF at a given event in one coordinate system then there is a timelike KVF at that event in all coordinate systems.

 

[Dale]

TrickyDicky, why don't we work through the following exercise. Please pick any spacetime with two coordinate systems which you believe illustrate your point in some region of the manifold. Then, let's take anyone timelike KVF in that region, write it in terms of both sets of coordinates, and calculate the Lie derivative and the norm in each set of coordinates.