# Gravitational time dilatation and curved spacetime - follow up

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PeterDonis
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2019 Award
parallel transporting
I should clarify this to forestall a question that might be coming from @pervect in view of the post of his that I just responded to. Since the ground observer's worldline is not a geodesic, the correct term for how the 4-momentum of emission of the light pulse is transported along that worldline is Fermi-Walker transport, not parallel transport. Or, more relevant to the current discussion, Lie transport (i.e., transport by the Lie derivative along the worldline), since that is what preserves the inner product of the 4-momentum with the Killing vector field.

I should clarify this to forestall a question that might be coming from @pervect in view of the post of his that I just responded to. Since the ground observer's worldline is not a geodesic, the correct term for how the 4-momentum of emission of the light pulse is transported along that worldline is Fermi-Walker transport, not parallel transport. Or, more relevant to the current discussion, Lie transport (i.e., transport by the Lie derivative along the worldline), since that is what preserves the inner product of the 4-momentum with the Killing vector field.
Thus 4-momentum of light pulse emission transported by Levi-Civita affine connection along observer wordlines at fixed $r$ (orbits of the flow of the killing vector $\partial_t$ in Schwarzschild coordinates) could be different from that due to Lie transport along the same observer worldlines ?

PeterDonis
Mentor
2019 Award
4-momentum of light pulse emission transported by Levi-Civita affine connection along observer wordlines at fixed $r$ (orbits of the flow of the killing vector $\partial_t$ in Schwarzschild coordinates) could be different from that due to Lie transport along the same observer worldlines ?
First, a clarification: if you have a non-geodesic worldline, "parallel transport" ("transport by the Levi-Civita connection") along it makes no sense. You have to take into account the path curvature of the worldline. Fermi-Walker transport is how you do that.

Second, "Lie transport" along an integral curve of a Killing vector field is the same as Fermi-Walker transport along that curve. The term "Lie transport" just makes clearer why it is relevant for this particular problem--heuristically, because it maintains the "geometric shape". Fermi-Walker transport in general makes no guarantee about that.

PAllen
2019 Award
First, a clarification: if you have a non-geodesic worldline, "parallel transport" ("transport by the Levi-Civita connection") along it makes no sense. You have to take into account the path curvature of the worldline. Fermi-Walker transport is how you do that.
Could you explain more? Parallel transport is well defined along any path. Specifically, transport around a closed loop (in the limit of ever smaller loops) is how curvature is defined.

PeterDonis
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2019 Award
Parallel transport is well defined along any path.
Yes, I was sloppy in my wording. Let me try to make it more precise. I was thinking of the discussion of Fermi-Walker transport in Chapter 7 of MTW.

Suppose we have an orthonormal tetrad $e_0, e_1, e_2, e_3$ of basis vectors at some event, such that the timelike vector $e_0$ coincides with the 4-velocity of some observer whose worldline passes through that event. We want to figure out how to transport that tetrad along the observer's worldline so that the transported $e_0$ remains coincident with the observer's 4-velocity everywhere along the worldline. Or, equivalently, $e_0$ starts out tangent to the worldline and we want to transport the tetrad so $e_0$ remains tangent to the worldline and the tetrad remains orthonormal.

If the worldline is a geodesic, parallel transport does the job.

If the worldline is not a geodesic, however, parallel transport does not do the job. The parallel transported $e_0$ will not remain tangent to the worldline. So we need to correct for that. We do that, as MTW describes, by adding to our transport law a Lorentz boost at each event that rotates, in spacetime, the parallel transported $e_0$ so that it is tangent to the worldline at that event. (At the original event, where $e_0$ was already tangent to the worldline, this boost just becomes the identity.) This boost will also rotate the spacelike vectors of the tetrad so that they remain orthogonal to $e_0$--parallel transport preserves orthogonality, so the parallel transported spacelike vectors will be orthogonal to the parallel transported $e_0$, and the Lorentz boost also preserves orthogonality, so the boosted vectors will still be mutually orthogonal. The result of this entire process is termed Fermi-Walker transport.

If we are talking about transporting vectors along a Killing flow, as in this discussion, Fermi-Walker transport is the transport law that is equivalent to Lie transport along the Killing flow. So that's the one we are interested in for this discussion.

transport around a closed loop (in the limit of ever smaller loops) is how curvature is defined.
As I understand it, curvature is defined in terms of parallel transport around a closed geodesic loop. You can't use an arbitrary loop. Since parallel transport in curved spacetime is path dependent, it would not make sense to allow a loop composed of arbitrary curves, since that would make curvature non-unique.

PAllen
2019 Award
Yes, I was sloppy in my wording. Let me try to make it more precise. I was thinking of the discussion of Fermi-Walker transport in Chapter 7 of MTW.

Suppose we have an orthonormal tetrad $e_0, e_1, e_2, e_3$ of basis vectors at some event, such that the timelike vector $e_0$ coincides with the 4-velocity of some observer whose worldline passes through that event. We want to figure out how to transport that tetrad along the observer's worldline so that the transported $e_0$ remains coincident with the observer's 4-velocity everywhere along the worldline. Or, equivalently, $e_0$ starts out tangent to the worldline and we want to transport the tetrad so $e_0$ remains tangent to the worldline and the tetrad remains orthonormal.

If the worldline is a geodesic, parallel transport does the job.

If the worldline is not a geodesic, however, parallel transport does not do the job. The parallel transported $e_0$ will not remain tangent to the worldline. So we need to correct for that. We do that, as MTW describes, by adding to our transport law a Lorentz boost at each event that rotates, in spacetime, the parallel transported $e_0$ so that it is tangent to the worldline at that event. (At the original event, where $e_0$ was already tangent to the worldline, this boost just becomes the identity.) This boost will also rotate the spacelike vectors of the tetrad so that they remain orthogonal to $e_0$--parallel transport preserves orthogonality, so the parallel transported spacelike vectors will be orthogonal to the parallel transported $e_0$, and the Lorentz boost also preserves orthogonality, so the boosted vectors will still be mutually orthogonal. The result of this entire process is termed Fermi-Walker transport.

If we are talking about transporting vectors along a Killing flow, as in this discussion, Fermi-Walker transport is the transport law that is equivalent to Lie transport along the Killing flow. So that's the one we are interested in for this discussion.
Fine, I summarize my understanding as:

Fermi-Walker transport preserves angle between vector and curve, and defines 'non-rotation'.

Parallel transport preserves direction and magnitude of the vector (magnitude only relevant given metric), relative to itself.

Both rely on a connection to define what vectors are the same at 'nearby' points.
As I understand it, curvature is defined in terms of parallel transport around a closed geodesic loop. You can't use an arbitrary loop. Since parallel transport in curved spacetime is path dependent, it would not make sense to allow a loop composed of arbitrary curves, since that would make curvature non-unique.
There is no such thing as a geodesic loop, in general. Geodesics very rarely form loops. A sequence of segments of different geodesics forming a loop has no special properties. Especially, note that there uncountably infinite choices for such a construction. It is only the limiting process that makes the result unique.

PeterDonis
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2019 Award
There is no such thing as a geodesic loop
To be clear, I didn't mean a closed loop formed by a single geodesic. I meant a closed loop formed by segments of multiple geodesics. The segments have to be geodesic segments because that is the only way to uniquely define a curve starting from a given event and with tangent vector at that event specified by a given vector. See, for example, MTW section 11.4.

PeterDonis
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2019 Award
See, for example, MTW section 11.4.
Ah, I just looked at MTW Box 11.7, which generalizes the treatment of section 11.4 to small closed curves of arbitrary shape. So I was wrong that the general definition of curvature is limited to a loop formed of geodesic segments.

PAllen
2019 Award
Ah, I just looked at MTW Box 11.7, which generalizes the treatment of section 11.4 to small closed curves of arbitrary shape. So I was wrong that the general definition of curvature is limited to a loop formed of geodesic segments.
Saved me from summarizing the treatment in Synge and Schild.

vanhees71
Gold Member
2019 Award
As I understand it, curvature is defined in terms of parallel transport around a closed geodesic loop. You can't use an arbitrary loop. Since parallel transport in curved spacetime is path dependent, it would not make sense to allow a loop composed of arbitrary curves, since that would make curvature non-unique.
The Riemann curvature tensor is geometrically defined by parallel transport around an arbitrary "inifinitesimal" closed curve in your manifold. You don't need to make a closed loop of geodesics.

The construction is as follows: Consider covariant vector components. Then the change of this vector when parallel transporting it along an infinitesimal closed loop $\delta C$ is
$$\Delta A_{\mu} = \int_{\delta C} \mathrm{d} x^{\alpha} {\Gamma^{\beta}}_{\mu \alpha} A_{\beta}.$$
Now use Stoke's integral theorem. With the surface-elements $\delta f^{\mu \nu}$ of a surface whose boundary $\delta C$ you get
$$\Delta A_{\mu} = \frac{1}{2} \left [\partial_{\gamma} \left ({\Gamma^{\beta}}_{\mu \alpha} \right )-\partial_{\alpha} \left ({\Gamma^{\beta}}_{\mu \gamma} \right) \right] \delta f^{\gamma \alpha}.$$
Now up to higher-order corrections we have
$$\partial_{\alpha} A_{\beta}={\Gamma^{\gamma}}_{\alpha \beta} A_{\gamma},$$
and after some algebra you get
$$\Delta A_{\mu}=\frac{1}{2} {R^{\alpha}}_{\mu \beta \gamma} A_{\alpha} \delta f^{\beta \gamma}$$
with
$${R^{\alpha}}_{\mu \beta \gamma} = \partial_{\beta} {\Gamma^{\alpha}}_{\mu \gamma} - \partial_{\gamma} {\Gamma^{\alpha}}_{\mu \beta} + {\Gamma^{\alpha}}_{\delta \beta} {\Gamma^{\delta}}_{\mu \gamma} - {\Gamma^{\alpha}}_{\delta \gamma} {\Gamma^{\delta}}_{\mu \beta},$$
which are the components of the Riemann curvature tensor.

PeterDonis
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2019 Award
You don't need to make a closed loop of geodesics.

pervect
Staff Emeritus
Going back to Fermi-Walker transport, I tend to think of it as the sort of transports that gyroscopes do. But, that view really only works for transporting vectors along the sort of curves that gyroscopes can follow, i.e. along timelike curves.

I suspect the OP want's a more general description of Fermi-Walker transport than that, one that could work along space-like curves. I don't have one to offer at this time, but there were some good posts on the topic by others.

In flat space-time, parallel transport in any inertial frame is just a matter of translating the vector in said frame. The curved space-time case is more interesting, but I don't think I need to rehash what others wrote there.

As far as the difference between Fermi-walker transport and parallel transport goes, I suggest comparing them in flat space time. Then if we restrict the transport to being along timelike curves (which is the way I think of it), Thomas precession of gyroscopes illustrates the difference between the two. In that context, transporting a vector in an inertial frame is just a matter of translating it, and Thomas precision shows that gyroscopes (and Fermi-Walker transport) obey a different transport law.

Wiki has an article on Thomas precession in 3-vector formalism, https://en.wikipedia.org/wiki/Thomas_precession#Statement. In the low velocity limit it's proportional to the cross product of the acceleration times the velocity, there are some additional multiplicative factors in the full relativistic case.

PeterDonis
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2019 Award
I suspect the OP want's a more general description of Fermi-Walker transport than that, one that could work along space-like curves.
You can apply the Fermi-Walker transport law along any spacelike curve; you just pick one of the spacelike vectors in the orthonormal tetrad to be the tangent vector to the spacelike curve. This case is not much discussed in the literature, but it works fine mathematically, since the Fermi-Walker transport law does not depend on the curve being timelike.

The case I'm not sure about is applying the Fermi-Walker transport law along a null curve. Obviously you can't choose an orthonormal tetrad in that case if you want to include the curve's tangent vector.

pervect
Staff Emeritus
You can apply the Fermi-Walker transport law along any spacelike curve; you just pick one of the spacelike vectors in the orthonormal tetrad to be the tangent vector to the spacelike curve. This case is not much discussed in the literature, but it works fine mathematically, since the Fermi-Walker transport law does not depend on the curve being timelike.

The case I'm not sure about is applying the Fermi-Walker transport law along a null curve. Obviously you can't choose an orthonormal tetrad in that case if you want to include the curve's tangent vector.
I imagine so - the formulas I am used to and looked up are expressed in terms of the acceleration and velocity along a time-like worldline. I would assume that one could use any affinely parameterized curve with suitable adjustments, but I have no intuition for what happens in that case.

The particular formula I use is MTW 6.14

$$\frac{d}{d\tau} v^a = \left( u^a a^b - u^b a^a \right) v_b \quad a^a = \frac{d}{d\tau} u^a$$

Here $v^a$ is the vector being transported, when the equation is satisfied, $v^a$ is fermi-walker transproted. $u^a$ is the unit time-like vector field that does the transporting - physically, it's just the 4-velocity (and 4-acceleration) of the path that the gyroscope takes, while $v^a$ is a vector that represents the spin-axis of the gyroscope.

While I am used to thinking of $u^a$ as a 4-velocity and $a^a$ as a 4-acceleration as above, I imagine it's not essential, $u^a$ would be interpreted as the tangent vector to the generalized curve, and $a^a$ would be it's derivative with respect to the affine parameter along the generalized curve. (I'd use s instead of $\tau$ to paramaterize a non-timelike curve, though of course that's just a matter of convention).

I also am not sure about potential normalization issues in the null case, I imagine that's your concern as well. In the timelike case, it's understood that $u^a$, the vector field representing the transport, is everywhere of unit length. I think it probably matters that it be of unit length.

Wiki has a more abstract treatment in terms of the Fermi derivative, rather than just writing the components of the transport equation, but that's not the approach I learned the topic from. Also, it's been a while since I've actually done anything like this.

In flat space-time, parallel transport in any inertial frame is just a matter of translating the vector in said frame. The curved space-time case is more interesting, but I don't think I need to rehash what others wrote there.
In that case (flat space-time in global Lorentzian coordinates) space-time is basically an affine space in affine coordinates thus parallel transporting a tangent vector (by Levi-Civita trivial connection) simply preserves its components.

In the context of this thread, IIUC, Lie transport along the flow of $\partial_t$ (Schwarzschild spacetime in Schwarzschild coordinates KVF) results the same as the Fermi-walker transport along it. Here we're interested at Lie transport because of its feature to preserve the 'shape' of the transported curve (light pulse paths) thus defining the notion "to be congruent"

PeterDonis
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2019 Award
I think it probably matters that it be of unit length.
Yes, I think it does, because IIRC the fact that $u^a$ and $a^a$ are orthogonal is used in the derivation of the formula you give. In the null case that orthogonality condition is no longer met.

vanhees71
Gold Member
2019 Award
But even for a null vector line you have
$$u_{\mu} u^{\mu}=0=\text{const}$$
and thus
$$\dot{u}_{\mu} u^{\mu}=0,$$
where the dot means a derivative wrt. an arbitrary parameter parametrizing the light-like worldline under consideration.

I'm not sure, whether one can define a congruence of "light-cone coordinates" by Fermi-Walker transport though (or a tetrad with two light-like and two real space-like vectros). In special relativity there's no problem to work with a basis consisting of two light-like and two space-like vectors. Of course these are not pseudo-Cartesian coordinates.

The only somewhat related thing I've heard about in GR is the use of Newton-Penrose tetrades, but they use four nullvectors with two of them complex. That's obviously not exactly the same as light-cone coordinates in SR.

PeterDonis
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2019 Award
even for a null vector line you have
$$u_{\mu} u^{\mu}=0=\text{const}$$
and thus
$$\dot{u}_{\mu} u^{\mu}=0$$
where the dot means a derivative wrt. an arbitrary parameter
Yes, you're right. I misstated the concern. The concern is that, for the null case, $\dot{u}$ must be tangent to the worldline. This obviously follows from what you wrote, quoted just above.

vanhees71
Gold Member
2019 Award
Maybe my notation was not well enough explained; $u^{\mu}$ is the tangent vector already. $\dot{u}^{\mu}$ is then something like the "acceleration", but for light-like curves $u^{\mu}$ is not a four-velocity in the usual sense anymore, and the world-line parameter can of course not be an intrinsic measure of time like proper time for time-like world lines (like the ones describing massive point-like classical particles). So it's not so clear, whether it makes sense to think about $a^{\mu}=\dot{u}^{\mu}$ as an "acceleration" in this case. Anyway, there are no massless classical point particles (even not in SR). So it's anyway an academic question. The one thing null geodesics are good for of course is as a model of "light beams" or rather propagation of em. waves using the eikonal approximation. The usual textbook shortcut is to talk about pointlike photons, which usually gives the right result (e.g., the light bending in gravitational fields) but a wrong qualitative picture, but that's another story.

PeterDonis
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2019 Award
Maybe my notation was not well enough explained
Your notation is fine; it's fairly standard to use the dot to denote a derivative with respect to the affine parameter along a curve.

I am simply pointing out that, for the null case, $\dot{u}$ must also be tangent to the null curve, as $u$ itself is; that follows immediately from $\dot{u}_\mu u^\mu = 0$. To put it another way, null vectors have the counterintuitive property that a vector that is orthogonal to a given null vector is also a multiple of it. Thus, we must have $\dot{u} = k u$, where $k$ is some function of the curve parameter (I don't think it needs to be constant along the curve). If we plug this into the Fermi derivative formula (using $\dot{u}$ instead of $a$), it makes the Fermi derivative vanish identically. So I don't think a useful notion of Fermi-Walker transport can be defined for null curves.

vanhees71
Gold Member
2019 Award
Sure, now I understand what you mean. Of course a vector $a$ that is "orthogonal" to a null vector $u$, $a^{\mu} u_{\mu}=g_{\mu \nu} a^{\mu} u^{\nu}$ can be either a null vector itself (than it's parallel to $u$, as you say) or it can be spacelike, depending on the specific light-like curve.

E.g., take the nullvector $(u^{\mu})=(1,0,0,1)$ in SR. Then any null vector $(a^{\mu})=(a^0,\vec{a})$ that is "orthogonal" to $u$ implies $a^0-a^3=0$ and thus $a^0=a^3$ and the from $(a^0)^2-\vec{a}^2=0$ that $a^1=a^2=0$ and thus $a=a^0 u$. But of course you can have two space-like directions that are "orthogonal" to $u$, e.g., $e_1=(0,1,0,0)$ and $e_2=(0,0,1,0)$. Light-cone coordinates are usually chosen by defining the basis consisting of the two light-like vectors $u_{\pm}=(1,0,0,\pm 1)$ and the two space-like vectors $e_1$ and $e_2$.

PeterDonis
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2019 Award
can be spacelike
Ah, that's right, there will always be a spacelike 2-surface orthogonal to the null vector.

depending on the specific light-like curve
I think there is a spacelike 2-surface orthogonal to any null curve; I don't think this depends on the specific curve.

You might want to study post #177 in that thread for a summary of the accumulated understanding developed int the thread.
Quoting first item in that post:
1) They posit a theory where the Minkowski metric is the observable metric for distance and time measurements.This is pretty clearly stated."
What do you mean with 'observable' Minkowski metric ? Basically that distance and time measurement are consistent with a Minkowski metric in which the (timelike) paths of observers at fixed height $z$ from ground are assumed to be geodesics of the underlying space-time ?

PAllen
2019 Award
Quoting first item in that post:

What do you mean with 'observable' Minkowski metric ? Basically that distance and time measurement are consistent with a Minkowski metric in which the (timelike) paths of observers at fixed height $z$ from ground are assumed to be geodesics of the underlying space-time ?
Yes.

Quoting third item in that post
3) They posit it is possible to set up global Lorentz frame physically using a described procedure.
Which is the procedure they posit to be used to setup that global Lorentz frame ?

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