In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.
Dalam matematika dan fisika, simbol Christoffel adalah deretan angka yang menggambarkan koneksi metrik.
In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric.
Dalam geometri diferensial, koneksi affine dapat didefinisikan tanpa mengacu pada metrik, dan banyak konsep tambahan berikut: transpor paralel, turunan kovarian, geodesik, dll.
However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor.
Namun, ketika metrik tersedia, konsep ini dapat langsung dikaitkan dengan "bentuk" manifold itu sendiri; bentuk itu ditentukan oleh bagaimana ruang singgung dilekatkan ke ruang kotangen oleh tensor metrik.
The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold.
Simbol Christoffel memberikan representasi konkret dari koneksi (pseudo-) geometri Riemannian dalam hal koordinat pada manifold.
For example, in Euclidean spaces, the Christoffel symbols describe how the local coordinate bases change from point to point.
Kemudian contohnya dalam ruang Euclidean, simbol Christoffel menggambarkan bagaimana basis koordinat lokal berubah dari titik ke titik.
The Christoffel symbols are named for Elwin Bruno Christoffel (1829–1900).
Simbol Christoffel dinamai oleh Elwin Bruno Christoffel.
1 2 g i m ( g m k , l + g m l , k − g k l , m ) , }_{kl}={\frac {1}{2}}g^{im}\left({\frac {\partial g_{mk}}{\partial x^{l}}}+{\frac {\partial g_{ml}}{\partial x^{k}}}-{\frac {\partial g_{kl}}{\partial x^{m}}}\right)={\frac {1}{2}}g^{im}\left(g_{mk,l}+g_{ml,k}-g_{kl,m}\right),} where (gjk) is the inverse of the matrix (gjk), defined as (using the Kronecker delta, and Einstein notation for summation) g j i g i kδ j k g_{ik}=\delta ^{j}{}_{k}} .
1 2 g i m ( g m k , l + g m l , k − g k l , m ) , }_{kl}={\frac {1}{2}}g^{im}\left({\frac {\partial g_{mk}}{\partial x^{l}}}+{\frac {\partial g_{ml}}{\partial x^{k}}}-{\frac {\partial g_{kl}}{\partial x^{m}}}\right)={\frac {1}{2}}g^{im}\left(g_{mk,l}+g_{ml,k}-g_{kl,m}\right),} di mana g i k } adalah invers metrik atau 1 g i k {g}}_{ik}} Simbol Christoffel sering digunakan dalam teori Relativitas umum Einstein, di mana ruang-waktu diwakili oleh manifold Lorentz 4-dimensi melengkung dengan koneksi Levi-Civita.
\Gamma _{cab}&={\frac {1}{2}}\left({\frac {\partial g_{ca}}{\partial x^{b}}}+{\frac {\partial g_{cb}}{\partial x^{a}}}-{\frac {\partial g_{ab}}{\partial x^{c}}}\right)\\&={\frac {1}{2}}\,\left(g_{ca,b}+g_{cb,a}-g_{ab,c}\right)\\&={\frac {1}{2}}\,\left(\partial _{b}g_{ca}+\partial _{a}g_{cb}-\partial _{c}g_{ab}\right)\,.\\\end{aligned}}} As an alternative notation one also finds Γ c a b
={\frac {1}{2}}\left({\frac {\partial g_{ca}}{\partial x^{b}}}+{\frac {\partial g_{cb}}{\partial x^{a}}}-{\frac {\partial g_{ab}}{\partial x^{c}}}\right)={\frac {1}{2}}\,\left(g_{ca,b}+g_{cb,a}-g_{ab,c}\right)={\frac {1}{2}}\,\left(\partial _{b}g_{ca}+\partial _{a}g_{cb}-\partial _{c}g_{ab}\right)\,.} simbol Christoffel kedua Γ i k l1 2 g i m ( ∂ g m k ∂ x l + ∂ g m l ∂ x k − ∂ g k l ∂ x m )
x ¯ i ∂ x m }^{i}}_{kl}={\frac {\partial {\bar {x}}^{i}}{\partial x^{m}}}\,{\frac {\partial x^{n}}{\partial {\bar {x}}^{k}}}\,{\frac {\partial x^{p}}{\partial {\bar {x}}^{l}}}\,{\Gamma ^{m}}_{np}+{\frac {\partial ^{2}x^{m}}{\partial {\bar {x}}^{k}\partial {\bar {x}}^{l}}}\,{\frac {\partial {\bar {x}}^{i}}{\partial x^{m}}}} where the overline denotes the Christoffel symbols in the x ¯ i }^{i}} coordinate system.
={\frac {1}{2}}\left({\frac {\partial g_{ca}}{\partial x^{b}}}+{\frac {\partial g_{cb}}{\partial x^{a}}}-{\frac {\partial g_{ab}}{\partial x^{c}}}\right)={\frac {1}{2}}\,\left(g_{ca,b}+g_{cb,a}-g_{ab,c}\right)={\frac {1}{2}}\,\left(\partial _{b}g_{ca}+\partial _{a}g_{cb}-\partial _{c}g_{ab}\right)\,.} simbol Christoffel kedua Γ i k l1 2 g i m ( ∂ g m k ∂ x l + ∂ g m l ∂ x k − ∂ g k l ∂ x m )
The Einstein field equations—which determine the geometry of spacetime in the presence of matter—contain the Ricci tensor, and so calculating the Christoffel symbols is essential.
Persamaan medan Einstein—yang menentukan geometri ruangwaktu dengan adanya materi—mengandung tensor Ricci, dan karenanya menghitung simbol Christoffel sangat penting.
Once the geometry is determined, the paths of particles and light beams are calculated by solving the geodesic equations in which the Christoffel symbols explicitly appear.
Setelah geometri ditentukan, jalur partikel dan berkas cahaya dihitung dengan memecahkan persamaan geodesik di mana simbol Christoffel muncul secara eksplisit.
{\ddot {x}}^{k}+{\frac {1}{2}}\left({\frac {\partial g_{ik}}{\partial x^{l}}}+{\frac {\partial g_{il}}{\partial x^{k}}}-{\frac {\partial g_{lk}}{\partial x^{i}}}\right){\dot {x}}^{l}{\dot {x}}^{k}=F_{i}.} Now multiplying by g i j } , we get x ¨ j + Γ j l k x ˙ l x ˙ kF j .
={\frac {1}{2}}\left({\frac {\partial g_{ca}}{\partial x^{b}}}+{\frac {\partial g_{cb}}{\partial x^{a}}}-{\frac {\partial g_{ab}}{\partial x^{c}}}\right)={\frac {1}{2}}\,\left(g_{ca,b}+g_{cb,a}-g_{ab,c}\right)={\frac {1}{2}}\,\left(\partial _{b}g_{ca}+\partial _{a}g_{cb}-\partial _{c}g_{ab}\right)\,.} simbol Christoffel kedua Γ i k l1 2 g i m ( ∂ g m k ∂ x l + ∂ g m l ∂ x k − ∂ g k l ∂ x m )