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2.3 The Lie derivative and spacetime symmetries

From the above discussion it should be evident that there are other ways to take derivatives in a curved spacetime. A particularly important tool for measuring changes in tensors from point to point in spacetime is the Lie derivative. It requires a vector field, but no connection, and is a more natural definition in the sense that it does not even require a metric. It yields a tensor of the same type and rank as the tensor on which the derivative operated (unlike the covariant derivative, which increases the rank by one). The Lie derivative is as important for Newtonian, non-relativistic fluids as for relativistic ones (a fact which needs to be continually emphasized as it has not yet permeated the fluid literature for chemists, engineers, and physicists). For instance, the classic papers on the Chandrasekhar–Friedman–Schutz instability [44Jump To The Next Citation Point45Jump To The Next Citation Point] in rotating stars are great illustrations of the use of the Lie derivative in Newtonian physics. We recommend the book by Schutz [104] for a complete discussion and derivation of the Lie derivative and its role in Newtonian fluid dynamics (see also the recent series of papers by Carter and Chamel [22Jump To The Next Citation Point23Jump To The Next Citation Point24Jump To The Next Citation Point]). We will adapt here the coordinate-based discussion of Schouten [99], as it may be more readily understood by those not well-versed in differential geometry.

In a first course on classical mechanics, when students encounter rotations, they are introduced to the idea of active and passive transformations. An active transformation would be to fix the origin and axis-orientations of a given coordinate system with respect to some external observer, and then move an object from one point to another point of the same coordinate system. A passive transformation would be to place an object so that it remains fixed with respect to some external observer, and then induce a rotation of the object with respect to a given coordinate system, by rotating the coordinate system itself with respect to the external observer. We will derive the Lie derivative of a vector by first performing an active transformation and then following this with a passive transformation and finding how the final vector differs from its original form. In the language of differential geometry, we will first “push-forward” the vector, and then subject it to a “pull-back”.

In the active, push-forward sense we imagine that there are two spacetime points connected by a smooth curve μ x (λ ). Let the first point be at λ = 0, and the second, nearby point at λ = ε, i.e. μ x (ε); that is,

xμ ≡ xμ(ε) ≈ xμ + εξμ, (36 ) ε 0
where xμ0 ≡ xμ(0) and
μ || ξμ = dx--|| (37 ) d λ |λ=0
is the tangent to the curve at λ = 0. In the passive, pull-back sense we imagine that the coordinate system itself is changed to -- -- xμ = xμ(x ν), but in the very special form
-- xμ = xμ − ε ξμ. (38 )
It is in this last step that the Lie derivative differs from the covariant derivative. In fact, if we insert Equation (36View Equation) into Equation (38View Equation) we find the result -μ μ xε = x0. This is called “Lie-dragging” of the coordinate frame, meaning that the coordinates at λ = 0 are carried along so that at λ = ε, and in the new coordinate system, the coordinate labels take the same numerical values.
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Figure 3: A schematic illustration of the Lie derivative. The coordinate system is dragged along with the flow, and one can imagine an observer “taking derivatives” as he/she moves with the flow (see the discussion in the text).

As an interesting aside it is worth noting that Arnold [10], only a little whimsically, refers to this construction as the “fisherman’s derivative”. He imagines a fisherman sitting in a boat on a river, “taking derivatives” as the boat moves with the current. Playing on this imagery, the covariant derivative is cast with the high-test Zebco parallel transport fishing pole, the Lie derivative with the Shimano, Lie-dragging ultra-light. Let us now see how Lie-dragging reels in vectors.

For some given vector field that takes values V μ(λ), say, along the curve, we write

V μ0 = V μ(0) (39 )
for the value of μ V at λ = 0 and
V με = V μ(ε) (40 )
for the value at λ = ε. Because the two points μ x0 and μ xε are infinitesimally close (ε ≪ 1) and we can thus write
| μ μ ν ∂V μ|| Vε ≈ V0 + ε ξ ---ν|| (41 ) ∂x λ=0
for the value of V μ at the nearby point and in the same coordinate system. However, in the new coordinate system at the nearby point, we find
-- ( -μ )|| μ|| V με = ∂-x-V ν || ≈ Vεμ− ε V0ν∂-ξ-|| . (42 ) ∂x ν |λ=ε ∂x ν|λ=0
The Lie derivative now is defined to be
--μ μ ℒξV μ = lim V-ε −-V-- ε→0 ε ν∂V-μ- ν ∂ξμ- = ξ ∂xν − V ∂xν = ξν∇ V μ − V ν∇ ξμ, (43 ) ν ν
where we have dropped the “0” subscript and the last equality follows easily by noting Γ ρμν = Γ ρνμ.

The Lie derivative of a covector A μ is easily obtained by acting on the scalar A μV μ for an arbitrary vector μ V:

μ μ μ ℒξA μV = V ℒ ξA μ + A μℒ ξV = Vμℒ A + A (ξν∇ V μ − V ν∇ ξμ). (44 ) ξ μ μ ν ν
But, because AμV μ is a scalar,
ℒ A Vμ = ξ ν∇ A V μ ξ μ ν ν μμ μ = ξ (V ∇ νAμ + A μ∇ νV ), (45 )
and thus
V μ(ℒ ξAμ − ξν∇ νA μ − Aν∇ μξν) = 0. (46 )
Since μ V is arbitrary,
ν ν ℒξA μ = ξ ∇ νAμ + A ν∇ μξ . (47 )

We introduced in Equation (32View Equation) the effect of parallel transport on vector components. By contrast, the Lie-dragging of a vector causes its components to change as

μ μ δVℒ = ℒξV ε. (48 )
We see that if ℒξV μ = 0, then the components of the vector do not change as the vector is Lie-dragged. Suppose now that V μ represents a vector field and that there exists a corresponding congruence of curves with tangent given by μ ξ. If the components of the vector field do not change under Lie-dragging we can show that this implies a symmetry, meaning that a coordinate system can be found such that the vector components will not depend on one of the coordinates. This is a potentially very powerful statement.

Let μ ξ represent the tangent to the curves drawn out by, say, the μ = a coordinate. Then we can write a x (λ) = λ which means

μ μ ξ = δ a. (49 )
If the Lie derivative of V μ with respect to ξν vanishes we find
∂V μ ∂ξ μ ξν ---ν-= V ν---ν = 0. (50 ) ∂x ∂x
Using this in Equation (41View Equation) implies Vεμ= Vμ0, that is to say, the vector field V μ(xν) does not depend on the xa coordinate. Generally speaking, every ξμ that exists that causes the Lie derivative of a vector (or higher rank tensors) to vanish represents a symmetry.

Let us take the spacetime metric gμν as an example. A spacetime symmetry can be represented by a generating vector field ξμ such that

ℒ ξgμν = ∇μ ξν + ∇ νξμ = 0. (51 )
This is known as Killing’s equation, and solutions to this equation are naturally referred to as Killing vectors. As a particular case, let us consider the class of stationary, axisymmetric, and asymptotically flat spacetimes. These are highly relevant in the present context since they capture the physics of rotating, equilibrium configurations. In other words, the geometries that result are among the most fundamental, and useful, for the relativistic astrophysics of spinning black holes and neutron stars. Stationary, axisymmetric, and asymptotically flat spacetimes are such that [14]
  1. there exists a Killing vector μ t that is timelike at spatial infinity;
  2. there exists a Killing vector φ μ that vanishes on a timelike 2-surface (called the axis of symmetry), is spacelike everywhere else, and whose orbits are closed curves; and
  3. asymptotic flatness means the scalar products tμtμ, φμ φμ, and tμφ μ tend to, respectively, − 1, + ∞, and 0 at spatial infinity.

These conditions imply [16] that a coordinate system exists where

μ μ t = (1,0, 0,0), φ = (0, 0,0,1). (52 )
So, for instance, the two Lie derivatives of the metric gμν, say, are such that
ℒ g = tσ∇ g + g ∇ tσ + g ∇ tσ t μν σ μσν μσ νσ σν μ = gμσ∂νt + gσν∂μt = 0, (53 )
and
σ σ σ ℒ φgμν = φ ∇ σgμν + gμσ∇ νφ + gσν∇ μφ = gμσ∂νφσ + gσν∂μφ σ = 0. (54 )

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