- #1

binbagsss

- 1,265

- 11

... via plugging in the Fundamental theorem of Riemmanian Geometry :

##\Gamma^u_{ab}=\frac{1}{2}g^{uc}(\partial_ag_{bc}+\partial_bg_{ca}-\partial_cg_{ab})##

Expanding out the covariant definition gives the geodesic equation as:

(1) ##\ddot{x^u}+\Gamma^u_{ab} x^a x^b =0 ##

(2) Lagrangian is given by ## \sqrt{g_{ab}\dot{x^a}\dot{x^b}}## but assuming affinely parameterised: ( ## \frac{dL}{ds}=0 ## ) then can treat the Lagrangian as ##g_{ab}\dot{x^a}\dot{x^b}## and E-L equations give geodesic.

Expanding out E-L equations I have: ##2g_{uv}\ddot{x^v} +2\partial_a(g_{uv})\dot{x^a}\dot{x^v}-\partial_ug_{ab}\dot{x^b}\dot{x^a}##

I believe i need to multiply christoffel symbol by ##g_{uc}## and expression (1) by ##g_{uc}## doing so and plugging into (1) gives me :

##g_{uc} \ddot{x^u}+2\dot{x^a}\dot{x^b}(\partial_a g_{bc}+\partial_bg_{ca}-\partial_cg_{ab}) ##

Now I see I can idenitify the middle term in brackets with the last term in (2) ,but, I don't know what to do now, I think I could rename indicies in either term 1 or 3 in the brackets to idenitty with term 2, but i can't rename in both terms as to me it looks like i'd need to do, to sort out the factor of two. even so the first term not in brackets looks factor of 2 out.

thanks

##\Gamma^u_{ab}=\frac{1}{2}g^{uc}(\partial_ag_{bc}+\partial_bg_{ca}-\partial_cg_{ab})##

Expanding out the covariant definition gives the geodesic equation as:

(1) ##\ddot{x^u}+\Gamma^u_{ab} x^a x^b =0 ##

(2) Lagrangian is given by ## \sqrt{g_{ab}\dot{x^a}\dot{x^b}}## but assuming affinely parameterised: ( ## \frac{dL}{ds}=0 ## ) then can treat the Lagrangian as ##g_{ab}\dot{x^a}\dot{x^b}## and E-L equations give geodesic.

Expanding out E-L equations I have: ##2g_{uv}\ddot{x^v} +2\partial_a(g_{uv})\dot{x^a}\dot{x^v}-\partial_ug_{ab}\dot{x^b}\dot{x^a}##

I believe i need to multiply christoffel symbol by ##g_{uc}## and expression (1) by ##g_{uc}## doing so and plugging into (1) gives me :

##g_{uc} \ddot{x^u}+2\dot{x^a}\dot{x^b}(\partial_a g_{bc}+\partial_bg_{ca}-\partial_cg_{ab}) ##

Now I see I can idenitify the middle term in brackets with the last term in (2) ,but, I don't know what to do now, I think I could rename indicies in either term 1 or 3 in the brackets to idenitty with term 2, but i can't rename in both terms as to me it looks like i'd need to do, to sort out the factor of two. even so the first term not in brackets looks factor of 2 out.

thanks

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