Synchronous frame

A reference frame in which the time coordinate defines proper time for all co-moving observers is called 'synchronous'. It is built by choosing some time-like hypersurface as an origin, such that has in every point a normal along the time line (lies inside the light cone with an apex in that point); all interval elements on this hypersurface are space-like. A family of geodesics normal to this hypersurface are drawn and defined as the time coordinates with a beginning at the hypersurface. d s 2 = g α β d x α d x β + 2 g 0 α d x 0 d x α + g 00 ( d x 0 ) 2 , {displaystyle ds^{2}=g_{alpha eta },dx^{alpha },dx^{eta }+2g_{0alpha },dx^{0},dx^{alpha }+g_{00}left(dx^{0} ight)^{2},}     (eq. 1) d x 0 ( 2 ) = 1 g 00 ( − g 0 α d x α + ( g 0 α g 0 β − g α β g 00 ) d x α d x β ) , {displaystyle dx^{0(2)}={frac {1}{g_{00}}}left(-g_{0alpha },dx^{alpha }+{sqrt {left(g_{0alpha }g_{0eta }-g_{alpha eta }g_{00} ight),dx^{alpha },dx^{eta }}} ight),}     (eq. 2) d l 2 = ( − g α β + g 0 α g 0 β g 00 ) d x α d x β . {displaystyle dl^{2}=left(-g_{alpha eta }+{frac {g_{0alpha }g_{0eta }}{g_{00}}} ight),dx^{alpha },dx^{eta }.}     (eq. 3) Δ x 0 = − g 0 α d x α g 00 ≡ g α d x α . {displaystyle Delta x^{0}=-{frac {g_{0alpha },dx^{alpha }}{g_{00}}}equiv g_{alpha },dx^{alpha }.}     (eq. 4) Δ x 0 = g 0 i d x i = 0 {displaystyle Delta x_{0}=g_{0i}dx^{i}=0}     (eq. 5) d l 2 = γ α β d x α d x β , {displaystyle dl^{2}=gamma _{alpha eta },dx^{alpha },dx^{eta },}     (eq. 6) γ α β = − g α β + g 0 α g 0 β g 00 {displaystyle gamma _{alpha eta }=-g_{alpha eta }+{frac {g_{0alpha }g_{0eta }}{g_{00}}}}     (eq. 7) g α β g β 0 + g α 0 g 00 = 0 , {displaystyle g^{alpha eta }g_{eta 0}+g^{alpha 0}g_{00}=0,}     (eqs. 8) − g α β γ β γ = δ γ α , {displaystyle -g^{alpha eta }gamma _{eta gamma }=delta _{gamma }^{alpha },}     (eq. 9) γ α β = − g α β . {displaystyle gamma ^{alpha eta }=-g^{alpha eta }.}     (eq. 10) − g = g 00 γ . {displaystyle -g=g_{00}gamma .}     (eq. 11) g α = − g 0 α g 00 . {displaystyle g_{alpha }=-{frac {g_{0alpha }}{g_{00}}}.}     (eq. 12) g α = γ α β g β = − g 0 α . {displaystyle g^{alpha }=gamma ^{alpha eta }g_{eta }=-g^{0alpha }.}     (eq. 13) g 00 = 1 g 00 − g α g α . {displaystyle g^{00}={frac {1}{g_{00}}}-g_{alpha }g^{alpha }.}     (eq. 14) g 00 = 1 , g 0 α = 0 , {displaystyle g_{00}=1,quad g_{0alpha }=0,}     (eq. 15) d s 2 = d t 2 − g α β d x α d x β , {displaystyle ds^{2}=dt^{2}-g_{alpha eta },dx^{alpha },dx^{eta },}     (eq. 16) γ α β = − g α β . {displaystyle gamma _{alpha eta }=-g_{alpha eta }.}     (eq. 17) g i k ∂ S ∂ x i ∂ S ∂ x k = 1 , {displaystyle g^{ik}{frac {partial S}{partial x^{i}}}{frac {partial S}{partial x^{k}}}=1,}     (eq. 18a) S = f ( ξ α , x i ) + A ( ξ α ) , {displaystyle S=fleft(xi ^{alpha },x^{i} ight)+Aleft(xi ^{alpha } ight),}     (eq. 18b) ∂ f ∂ ξ α = − ∂ A ∂ ξ α . {displaystyle {frac {partial f}{partial xi ^{alpha }}}=-{frac {partial A}{partial xi ^{alpha }}}.}     (eq. 18c) x i = x ´ i + ξ i ( x ´ ) . {displaystyle x^{i}={acute {x}}^{i}+xi ^{i}({acute {x}}).}     (eq. 18) d s 2 = g i k ( x ) d x i d x k = g i k ( n e w ) ( x ´ ) d x ´ i d x ´ k , {displaystyle ds^{2}=g_{ik}(x)dx^{i}dx^{k}=g_{ik}^{(new)}({acute {x}})d{acute {x}}^{i}d{acute {x}}^{k},}     (eq. 19) g i k ( n e w ) ( x ´ ) = g i k ( x ´ ) + g i l ( x ´ ) ∂ ξ l ( x ´ ) ∂ x ´ k + g k l ( x ´ ) ∂ ξ l ( x ´ ) ∂ x ´ i + ∂ g i k ( x ´ ) ∂ x ´ l ξ l ( x ´ ) . {displaystyle g_{ik}^{(new)}({acute {x}})=g_{ik}({acute {x}})+g_{il}({acute {x}}){frac {partial xi ^{l}({acute {x}})}{partial {acute {x}}^{k}}}+g_{kl}({acute {x}}){frac {partial xi ^{l}({acute {x}})}{partial {acute {x}}^{i}}}+{frac {partial g_{ik}({acute {x}})}{partial {acute {x}}^{l}}}xi ^{l}({acute {x}}).}     (eq. 20) ∂ ξ 0 ( x ´ ) ∂ x ´ = 0 , g α β ( x ´ ) ∂ ξ β ( x ´ ) ∂ t ´ − ∂ ξ 0 ( x ´ ) ∂ x ´ α = 0. {displaystyle {frac {partial xi ^{0}({acute {x}})}{partial {acute {x}}}}=0,quad g_{alpha eta }({acute {x}}){frac {partial xi ^{eta }({acute {x}})}{partial {acute {t}}}}-{frac {partial xi ^{0}({acute {x}})}{partial {acute {x}}^{alpha }}}=0.}     (eq. 21) ξ 0 = f 0 ( x ´ 1 , x ´ 2 , x ´ 3 ) , ξ α = ∫ g α β ( x ´ ) ∂ f 0 ( x ´ 1 , x ´ 2 , x ´ 3 ) ∂ x ´ β d x ´ 0 + f α ( x ´ 1 , x ´ 2 , x ´ 3 ) , {displaystyle xi ^{0}=f^{0}left({acute {x}}^{1},{acute {x}}^{2},{acute {x}}^{3} ight),quad xi ^{alpha }=int {g^{alpha eta }({acute {x}}){frac {partial f^{0}left({acute {x}}^{1},{acute {x}}^{2},{acute {x}}^{3} ight)}{partial {acute {x}}^{eta }}}d{acute {x}}^{0}}+f^{alpha }left({acute {x}}^{1},{acute {x}}^{2},{acute {x}}^{3} ight),}     (eq. 22) ϰ α β = ∂ γ α β ∂ t {displaystyle varkappa _{alpha eta }={frac {partial gamma _{alpha eta }}{partial t}}}     (eq. 23) ϰ α α = γ α β ∂ γ α β ∂ t = ∂ ∂ t ln ⁡ ( γ ) . {displaystyle varkappa _{alpha }^{alpha }=gamma ^{alpha eta }{frac {partial gamma _{alpha eta }}{partial t}}={frac {partial }{partial t}}ln {(gamma )}.}     (eq. 24) Γ 00 0 = Γ 00 α = Γ 0 α 0 = 0 , Γ α β 0 = 1 2 ϰ α β , Γ 0 β α = 1 2 ϰ β α , Γ α β γ = λ α β γ {displaystyle Gamma _{00}^{0}=Gamma _{00}^{alpha }=Gamma _{0alpha }^{0}=0,quad Gamma _{alpha eta }^{0}={frac {1}{2}}varkappa _{alpha eta },quad Gamma _{0eta }^{alpha }={frac {1}{2}}varkappa _{eta }^{alpha },quad Gamma _{alpha eta }^{gamma }=lambda _{alpha eta }^{gamma }}     (eq. 25) λ α β γ = 1 2 γ γ μ ( γ μ α , β + γ μ β , β − γ α β , μ ) , {displaystyle lambda _{alpha eta }^{gamma }={frac {1}{2}}gamma ^{gamma mu }left(gamma _{mu alpha ,eta }+gamma _{mu eta ,eta }-gamma _{alpha eta ,mu } ight),}     (eq. 26) R 0 0 = − 1 2 ϰ ˙ − 1 4 ϰ α β ϰ β α , {displaystyle R_{0}^{0}=-{frac {1}{2}}{dot {varkappa }}-{frac {1}{4}}varkappa _{alpha }^{eta }varkappa _{eta }^{alpha },}     (eq. 27) R α 0 = 1 2 ( ϰ α ; β β − ϰ , α ) , {displaystyle R_{alpha }^{0}={frac {1}{2}}left(varkappa _{alpha ;eta }^{eta }-varkappa _{,alpha } ight),}     (eq. 28) R α β = − 1 2 γ ( γ ϰ α β ) ˙ − P α β . {displaystyle R_{alpha }^{eta }=-{frac {1}{2{sqrt {gamma }}}}{dot {left({sqrt {gamma }}varkappa _{alpha }^{eta } ight)}}-P_{alpha }^{eta }.}     (eq. 29) P α β = γ β γ P γ α , P α β = λ α β , γ γ − λ γ α , β γ + λ α β γ λ γ μ μ − λ α μ γ λ β γ μ . {displaystyle P_{alpha }^{eta }=gamma ^{eta gamma }P_{gamma alpha },quad P_{alpha eta }=lambda _{alpha eta ,gamma }^{gamma }-lambda _{gamma alpha ,eta }^{gamma }+lambda _{alpha eta }^{gamma }lambda _{gamma mu }^{mu }-lambda _{alpha mu }^{gamma }lambda _{eta gamma }^{mu }.}     (eq. 30) R 0 0 = − 1 2 ϰ ˙ − 1 4 ϰ α β ϰ β α = 8 π k ( T 0 0 − 1 2 T ) , {displaystyle R_{0}^{0}=-{frac {1}{2}}{dot {varkappa }}-{frac {1}{4}}varkappa _{alpha }^{eta }varkappa _{eta }^{alpha }=8pi kleft(T_{0}^{0}-{frac {1}{2}}T ight),}     (eq. 31) R α 0 = 1 2 ( ϰ α ; β β − ϰ , α ) = 8 π k T α 0 , {displaystyle R_{alpha }^{0}={frac {1}{2}}left(varkappa _{alpha ;eta }^{eta }-varkappa _{,alpha } ight)=8pi kT_{alpha }^{0},}     (eq. 32) R α β = − 1 2 γ ( γ ϰ α β ) ˙ − P α β = 8 π k ( T α β − 1 2 δ α β T ) . {displaystyle R_{alpha }^{eta }=-{frac {1}{2{sqrt {gamma }}}}{dot {left({sqrt {gamma }}varkappa _{alpha }^{eta } ight)}}-P_{alpha }^{eta }=8pi kleft(T_{alpha }^{eta }-{frac {1}{2}}delta _{alpha }^{eta }T ight).}     (eq. 33) A reference frame in which the time coordinate defines proper time for all co-moving observers is called 'synchronous'. It is built by choosing some time-like hypersurface as an origin, such that has in every point a normal along the time line (lies inside the light cone with an apex in that point); all interval elements on this hypersurface are space-like. A family of geodesics normal to this hypersurface are drawn and defined as the time coordinates with a beginning at the hypersurface.