The speed of light c and Planck constant ℏ are often seen to clutter the equations, so they are therefore often expressed in natural units where cℏ1 .
Dengan kecepatan cahaya c and konstanta planck ℏ dan dengan menggunakan kesepakatan satuan di mana cℏ1 .
0 , -{\frac {\ m^{2}c^{2}}{\ \hbar ^{2}}}\ \right]\ \psi (\ \mathbf {r} \ )
=-g^{\mu \nu }\nabla _{\mu }(\partial _{\nu }\psi )+{\dfrac {m^{2}c^{2}}{\hbar ^{2}}}\psi
)^{2}c^{2}+m^{2}c^{4}}}\,\psii\hbar {\frac {\partial }{\partial t}}\psi .}
=-g^{\mu \nu }\nabla _{\mu }(\partial _{\nu }\psi )+{\dfrac {m^{2}c^{2}}{\hbar ^{2}}}\psi
=\left(i\hbar {\frac {\partial }{\partial t}}\right)^{2}\psi ,} which simplifies to − ℏ 2 c 2 ∇ 2 ψ + m 2 c 4 ψ− ℏ 2 ∂ 2 ∂ t 2 ψ . c^{2}\mathbf {\nabla } ^{2}\psi +m^{2}c^{4}\psi-\hbar ^{2}{\frac {\partial ^{2}}{\partial t^{2}}}\psi .}
+ m 2 c 2 ℏ 2 ψ0 , {\sqrt {-g}}}\partial _{\mu }\left(g^{\mu \nu }{\sqrt {-g}}\partial _{\nu }\psi \right)+{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi0,} di mana gαβ adalah invers metrik dari tensor metrik, g adalah determinan dari tensor metrik, ∇μ adalah turunan kovarian, dan Γσμν adalah Simbol Christoffel Klein-Gordon equation (English Wikipedia)
Rearranging terms yields 1 c 2 ∂ 2 ∂ t 2 ψ − ∇ 2 ψ + m 2 c 2 ℏ 2 ψ0. {c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi -\mathbf {\nabla } ^{2}\psi +{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi0.} Since all reference to imaginary numbers has been eliminated from this equation, it can be applied to fields that are real-valued, as well as those that have complex values.
+ m 2 c 2 ℏ 2 ψ0 , {\sqrt {-g}}}\partial _{\mu }\left(g^{\mu \nu }{\sqrt {-g}}\partial _{\nu }\psi \right)+{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi0,} di mana gαβ adalah invers metrik dari tensor metrik, g adalah determinan dari tensor metrik, ∇μ adalah turunan kovarian, dan Γσμν adalah Simbol Christoffel Klein-Gordon equation (English Wikipedia)
In general relativity, we include the effect of gravity by replacing partial derivatives with covariant derivatives, and the Klein–Gordon equation becomes (in the mostly pluses signature) 0− g μ ν ∇ μ ∇ ν ψ + m 2 c 2 ℏ 2 ψ− g μ ν ∇ μ ( ∂ ν ψ )
Dalam relativitas umum, kami memasukkan efek gravitasi dengan mengganti parsial dengan turunan kovarian, dan persamaan Klein–Gordon menjadi (dalam tanda sebagian besar plus) 0− g μ ν ∇ μ ∇ ν ψ + m 2 c 2 ℏ 2 ψ− g μ ν ∇ μ ( ∂ ν ψ )
\\&=-g^{\mu \nu }\partial _{\mu }\partial _{\nu }\psi +g^{\mu \nu }\Gamma ^{\sigma }{}_{\mu \nu }\partial _{\sigma }\psi +{\dfrac {m^{2}c^{2}}{\hbar ^{2}}}\psi ,\end{aligned}}} or equivalently, − 1 − g ∂ μ ( g μ ν − g ∂ ν ψ )
\\&=-g^{\mu \nu }\partial _{\mu }\partial _{\nu }\psi +g^{\mu \nu }\Gamma ^{\sigma }{}_{\mu \nu }\partial _{\sigma }\psi +{\dfrac {m^{2}c^{2}}{\hbar ^{2}}}\psi ,\end{aligned}}} Atau bisa ditulis dengan, − 1 − g ∂ μ ( g μ ν − g ∂ ν ψ )
+ m 2 c 2 ℏ 2 ψ0 , {\sqrt {-g}}}\partial _{\mu }\left(g^{\mu \nu }{\sqrt {-g}}\partial _{\nu }\psi \right)+{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi0,} where gαβ is the inverse of the metric tensor that is the gravitational potential field, g is the determinant of the metric tensor, ∇μ is the covariant derivative, and Γσμν is the Christoffel symbol that is the gravitational force field.
+ m 2 c 2 ℏ 2 ψ0 , {\sqrt {-g}}}\partial _{\mu }\left(g^{\mu \nu }{\sqrt {-g}}\partial _{\nu }\psi \right)+{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi0,} di mana gαβ adalah invers metrik dari tensor metrik, g adalah determinan dari tensor metrik, ∇μ adalah turunan kovarian, dan Γσμν adalah Simbol Christoffel Klein-Gordon equation (English Wikipedia)