Cinco pilares de la arquitectura sostenible

Let {xα}be an arbitrary given coordinate system and{xα}the one which is desired: it reduces to the inertial system at a certain fixed pointP. (A point in this four-dimensional manifold is, of course, an event.) Then there is some relation

xα =xα(xμ), (6.21)

α_{μ =∂x}α_/∂xμ

. (6.22)

Expanding α_μ in a Taylor series about P (whose coordinates are xμ₀) gives the transformation at an arbitrary pointxnearP:

αμ(x)=α_μ(P)+(xγ−xγ₀)∂ α_μ ∂xγ (P) +1 2(x γ −xγ₀)(xλ−xλ0) ∂2α μ ∂xλ_∂xγ(P)+ · · ·, =αμ|_P+(xγ−xγ₀) ∂ 2_xα ∂xγ∂xμ ** ** P +1 2(x γ −xγ₀)(xλ−xλ0) ∂3_xα ∂xλ_∂xγ∂xμ ** ** P+ · · ·. (6.23) Expanding the metric in the same way gives

g_αβ(x)=g_αβ|_P+(xγ−xγ₀) ∂gαβ ∂xγ ** ** P +1 2(x γ −xγ₀)(xλ−xλ₀) ∂ 2_g αβ ∂xλ∂xγ ** ** * P + · · ·. (6.24) We put these into the transformation,

guν =αμβνgαβ, (6.25) to obtain g_μν(x)=α_μ|_Pβ_ν|_Pg_αβ|_P +(xγ−xγ₀)[α_μ|_Pβ_ν|_Pg_αβ,γ|P +αμ|_Pg_αβ|_P∂2xβ/∂xγ∂xν|_P +βν|_Pgαβ|P∂2xα/∂xγ∂xμ|_P] +1 2(x γ −xγ₀)(xλ −xλ₀)[· · ·]. (6.26) Now, we do not know the transformation, Eq. (6.21), but we can define it by its Taylor expansion. Let us count the number of free variables we have for this purpose. The matrix α_μ|

P has 16 numbers, all of which are freely specifiable. The array{∂2xα/∂xγ∂xμ|_P} has 4×10=40 free numbers (not 4×4×4, since it is symmetric in γ and μ). The array{∂3xα/∂xλ∂xγ∂xμ|_P}has 4×20=80 free variables, since symmetry onall

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rearrangements of λ,γ and μ gives only 20 independent arrangements (the general expression for three indices isn(n+1)(n+2)/3!, wherenis the number of values each index can take, four in our case). On the other hand,g_αβ|_P,g_αβ,γ|P andgαβ,γμ|P are all given initially. They have, respectively, 10, 10×4=40, and 10×10=100 independent numbers for a fully general metric. The first question is, can we satisfy Eq. (6.4),

g_μν|_P =η_μν? (6.27)

This can be written as

ημ_ν =α_μ|_Pβ_ν|_Pg_αβ|_P. (6.28) By symmetry, these are ten equations, which for general matrices are independent. To satisfy them we have 16 free values inα_μ|_P. The equations can indeed, therefore, be sat- isfied, leaving six elements ofα_μ|_Punspecified. These six correspond to the six degrees of freedom in the Lorentz transformations that preserve the form of the metricη_μν. That is, we can boost by a velocityv (three free parameters) or rotate by an angleθ around a direction defined by two other angles. These add up to six degrees of freedom inα_μ|_P

that leave the local inertial frame inertial.

The next question is, can we choose the 40 free numbers∂α_μ/∂xγ|_Pin Eq. (6.26) in such a way as to satisfy the 40 independent equations, Eq. (6.5),

g_αβ,_μ|P =0? (6.29)

Since 40 equals 40, the answer is yes, just barely. Given the matrixα_μ|_P, there is one and only one way to arrange the coordinates nearPsuch thatα_μ,γ|Phas the right values to makeg_αβ,μ|P=0. So there is no extra freedom other than that with which to make local Lorentz transformations.

The final question is, can we make this work at higher order? Can we find 80 numbers α_{μ,γλ|P which can make the 100 numbersgαβ,μλ|P =0? The answer, since 80<}

100, is no. There are, in the general metric, 20 ‘degrees of freedom’ among the second derivativesg_αβ,μλ|P. Since 100−80=20, there will be in general 20 components that cannot be made to vanish.

Therefore we see that a general metric is characterized at any pointPnot so much by its value atP(which can always be made to beη_αβ), nor by its first derivatives there (which can be made zero), but by the 20 second derivatives there which in general cannot be made to vanish. These 20 numbers will be seen to be the independent components of a tensor which represents the curvature; this we shall show later. In aflatspace, of course, all 20 vanish. In a general space they do not.

6.3 C o v a r i a n t d i f f e r e n t i a t i o n

We now look at the subject of differentiation. By definition, the derivative of a vector field involves the difference between vectors at two different points (in the limit as the points come together). In a curved space the notion of the difference between vectors at different

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points must be handled with care, since in between the points the space is curved and the idea that vectors at the two points might point in the ‘same’ direction is fuzzy. However, the local flatness of the Riemannian manifold helps us out. We only need to compare vectors in the limit as they get infinitesimally close together, and we know that we can construct a coordinate system at any point which is as close to being flat as we would like in this same limit. So in a small region the manifold looks flat, and it is then natural to say that the derivative of a vector whose components are constant in this coordinate system is zero at that point. In particular, we say that the derivatives of the basis vectors of a locally inertial coordinate system are zero atP.

Let us emphasize that this is adefinitionof the covariant derivative. For us, its justifica- tion is in the physics: the local inertial frame is a frame in which everything is locally like SR, and in SR the derivatives of these basis vectors are zero. This definition immediately leads to the fact that in these coordinates at this point, the covariant derivative of a vector has components given by the partial derivatives of the components (that is, the Christoffel symbols vanish):

Vα:β =Vα.β atPin this frame. (6.30) This is of course also true for any other tensor, including the metric:

g_αβ;γ =gαβ,γ =0 atP.

(The second equality is just Eq. (6.5).) Now, the equationg_αβ;γ =0 is true in one frame (the locally inertial one), and is a valid tensor equation; therefore it is true inanybasis:

g_αβ:γ =0 in any basis. (6.31) This is a very important result, and comes directly from our definition of the covariant derivative. Recalling §5.4, we see thatifwe haveμ_αβ=μ_βα, then Eq. (6.31) leads to Eq. (5.75) foranymetric:

αμν = 1

2g

αβ_(g

βμ,ν+gβν,μ−gμν,β). (6.32)

It is left to Exer. 5, § 6.9, to demonstrate, by repeating the flat-space argument now in the locally inertial frame, thatμ_βαis indeed symmetric in any coordinate system, so that Eq. (6.32) is correct in any coordinates. We assumed at the start that atPin a locally inertial frame,α_μν =0. But, importantly, the derivatives ofα_μν atP in this frame are not all zero generally, since they involveg_αβ,γ μ. This means that even though coordinates can be found in whichαμν=0 at a point, these symbols do not generally vanish elsewhere. This differs from flat space, where a coordinate system exists in whichα_μν =0 everywhere. So we can see that at any given point, the difference between a general manifold and a flat one manifests itself in the derivatives of the Christoffel symbols.

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Eq. (6.32) means that, giveng_αβ, we can calculateα_μνeverywhere. We can therefore calculate all covariant derivatives, giveng. To review the formulas:

Vα;β =Vα,β+αμβVμ, (6.33)

Pα;β =Pα,β−μαβPμ, (6.34) Tαβ;γ =Tαβ,γ +αμγTμβ+βμγTαμ. (6.35)