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Christoffel Symbol of the Second Kind -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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Christoffel symbols of the second kind are the second type of tensor-like object derived from a Riemannian metric g which is used to study the geometry of the metric. Christoffel symbols of the second kind are variously denoted as {m; i j} (Walton 1967) or Gamma^m_(ij) (Misner et al. 1973, Arfken 1985). They are also known as affine connections (Weinberg 1972, p. 71) or connection coefficients (Misner et al. 1973, p. 210).

Unfortunately, there are two different definitions of the Christoffel symbol of the second kind.

Arfken (1985, p. 161) defines

where partial/partialx is a partial derivative, g^(km) is the metric tensor,

epsilon_i=(partialr)/(partialq^i)

(4)

where r is the radius vector, and

epsilon^i=g^(ij)epsilon_j.

(5)

Therefore, for an orthogonal curvilinear coordinate system, by this definition,

Gamma^m_(ij)=1/(g_(mm))epsilon_m·(partial^2r)/(partialq^jpartialq^i).

(6)

The symmetry of definition (6) means that

Gamma^k_(ij)=Gamma^k_(ji)

(7)

(Walton 1967).

This Christoffel symbol of the second kind is related to the Christoffel symbol of the first kind [bc,d] by

Gamma^a_(bc)=g^(ad)[bc,d].

(8)

Walton (1967) lists Christoffel symbols of the second kind for the 12 basic orthogonal coordinate systems.

A different definition of Christoffel symbols of the second kind is given by

Gamma^k_(ij)=u_k^^·(del _ju_i^^)

(9)

(Misner et al. 1973, p. 209), where del _j denotes a gradient. Note that this kind of Christoffel symbol is not symmetric in i and j.

Christoffel symbols of the second kind are not tensors, but have tensor-like contravariant and covariant indices. Christoffel symbols of the second kind also do not transform as tensors. In fact, changing coordinates from x_1,...,x_n to y_1,...,y_n gives

Gamma^k_(ij)^'=sum(partial^2x_l)/(partialy_ipartialy_j)(partialy_k)/(partialx_l)+sumGamma^t_(rs)(partialx_r)/(partialy_i)(partialx_s)/(partialy_j)(partialy_k)/(partialx_t).

(10)

However, a fully covariant Christoffel symbol of the second kind is given by

Gamma_(alphabetagamma)=1/2(g_(alphabeta,gamma)+g_(alphagamma,beta)-g_(betagamma,alpha)+c_(alphabetagamma)+c_(alphagammabeta)-c_(betagammaalpha))

(11)

(Misner et al. 1973, p. 210), where the gs are the metric tensors, the cs are commutation coefficients, and the commas indicate the comma derivative. In an orthonormal basis, g_(alphabeta,gamma)=0 and g_(mugamma)=delta_(mugamma), so

Gamma_(alphabetagamma)=Gamma^mu_(alphabeta)g_(mugamma)=Gamma_(alphabeta)^mu=1/2(c_(alphabetagamma)+c_(alphagammabeta)-c_(betagammaalpha))

(12)

and

For tensors of tensor rank 3, the Christoffel symbols of the second kind may be concisely summarized in matrix form:

Gamma^l=[Gamma^l_(ii) Gamma^l_(ij) Gamma^l_(ik); Gamma^l_(ji) Gamma^l_(jj) Gamma^l_(jk); Gamma^l_(ki) Gamma^l_(kj) Gamma^l_(kk)].

(19)

The Christoffel symbols are given in terms of the coefficients of the first fundamental form E, F, and G by

and Gamma^1_(21)=Gamma^1_(12) and Gamma^2_(21)=Gamma^2_(12). If F=0, the Christoffel symbols of the second kind simplify to

(Gray 1997).

The following relationships hold between the Christoffel symbols of the second kind and coefficients of the first fundamental form,

(Gray 1997).

For a surface given in Monge's form z=F(x,y),

Gamma^k_(ij)=(z_(ij)z_k)/(1+z_1^2+z_2^2).

(40)

Christoffel symbols of the second kind arise in the computation of geodesics. The geodesic equation of free motion is

dtau^2=-eta_(alphabeta)dxi^alphadxi^beta,

(41)

or

(d^2xi^alpha)/(dtau^2)=0.

(42)

Expanding,

d/(dtau)((partialxi^alpha)/(partialx^mu)(dx^mu)/(dtau))=(partialxi^alpha)/(partialx^mu)(d^2x^mu)/(dtau^2)+(partial^2xi^alpha)/(partialx^mupartialx^nu)(dx^mu)/(dtau)(dx^nu)/(dtau)=0

(43)

(partialxi^alpha)/(partialx^mu)(d^2x^mu)/(dtau^2)(partialx^lambda)/(partialxi^alpha)+(partial^2xi^alpha)/(partialx^mupartialx^nu)(dx^mu)/(dtau)(dx^nu)/(dtau)(partialx^lambda)/(partialxi^alpha)=0.

(44)

But

(partialxi^alpha)/(partialx^nu)(partialx^lambda)/(partialxi^alpha)=delta_mu^lambda,

(45)

so

delta_mu^lambda(d^2x^mu)/(dtau^2)+((partial^2xi^alpha)/(partialx^mupartialx^nu)(partialx^lambda)/(partialxi^alpha))(dx^mu)/(dtau)(dx^nu)/(dtau)=(d^2x^lambda)/(dtau^2)+Gamma^lambda_(munu)(dx^mu)/(dtau)(dx^nu)/(dtau),

(46)

where

Gamma^lambda_(munu)=(partial^2xi^alpha)/(partialx^mupartialx^nu)(partialx^lambda)/(partialxi^alpha).

(47)

See also

Cartan Torsion Coefficient, Christoffel Symbol, Christoffel Symbol of the First Kind, Comma Derivative, Commutation Coefficient, Covariant Derivative, Gauss Equations, Tensor

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 160-167, 1985.Gray, A. "Christoffel Symbols." §22.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 509-513, 1997.Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. San Francisco: W. H. Freeman, 1973.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 47-48, 1953.Sternberg, S. Differential Geometry. New York: Chelsea, p. 354, 1983.Walton, J. J. "Tensor Calculations on Computer: Appendix." Comm. ACM 10, 183-186, 1967.Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, 1972.

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Christoffel Symbol of the Second Kind

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Weisstein, Eric W. "Christoffel Symbol of the Second Kind." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ChristoffelSymboloftheSecondKind.html

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