- #1
Tursinbay
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General physical perturbations of string is derived by A.Larsen and V.Frolov (arXiv:hep-th/9303001v1 1March 1993).
An arbitrary string configuration is in 4-dimensional gravitational background. Starting point is Polyakov action
$$ S = \int d \tau d\sigma \sqrt {-h} h^{AB} G_{AB}$$.
Here is ##h_{AB}## is internal metric with determinant ##h##. ##G_{AB}## is the induced metric on the world-sheet:
$$G_{AB}=g_{ \mu \nu} \frac {\partial x^{\mu}} {\partial \xi^A} \frac {\partial x^{\nu}} {\partial \xi^B} = g_{ \mu \nu} x^{\mu}_{,A} x^{\nu}_{,B}$$
Internal metric is two dimensional metric of world-sheet coordinates ##\xi^A## (A=0,1), ## (\xi^0, \xi^1)=(\tau, \sigma) ##. ##x^{\mu} (\mu=0,1,2,3) ## are the spacetime coordinates.
Variations of the action with respect to ## \delta x^{\mu} ## and ## \delta h_{AB} ## is
$$ \delta S = \int d \tau d \sigma \sqrt {-h} \left( [ \frac 1 2 h^{AB} G^C_{ C} - G^{AB}] \delta h_{AB} - 2g_{\mu \nu} [\Box x^{\nu} + h^{AB} \Gamma^{\nu}_{ \rho \sigma} x^{\rho}_{,A}x^{\sigma}_{,B}] \delta x^{\mu} \right)$$
Here ##G^C_{ C}=h^{BC}G_{BC}## is the trace of induced metric on the world-sheet and ##\Box ## is the d'Alambertian:
$$ \Box = \frac {1} {\sqrt{-h}} \partial_A(\sqrt{-h} h^{AB} \partial_B). $$
From ## \delta S## is derived the equations of motion as usual
$$\frac 1 2 h^{AB} G^C_{ C} - G^{AB}=0$$
$$\Box x^{\nu} + h^{AB} \Gamma^{\nu}_{ \rho \sigma} x^{\rho}_{,A}x^{\sigma}_{,B}=0. $$
I got the first variations exactly as in paper (but with plus sign between the variations ## \delta h_{AB}## and ## \delta x^{\mu}##). But, equations of perturbations are obtained from the second variations of action and the problem starts here.
One more variation from the first variation is taken. And before doing this is done the following assumptions.
1. ## x^{\mu}=x^{\mu}(\xi^A)## is the solution of the equations of motion.
2. Introduced 2 vectors ##n^{\mu}_R (R=2,3)## normal to the surface of the string world-sheet:
$$ g_{\mu \nu} n^{\mu}_R n^{\nu}_S=\delta_{RS}, \\ g_{\mu \nu}x^{\mu}_{,A} n^{\nu}_R =0. $$
3. The general perturbation ## \delta x^{\mu}## can be composed as: $$\delta x^{\mu}=\delta x^R n^{\mu}_R+\delta x^A x^{\mu}_{,A} $$
The variations ##\delta x^A x^{\mu}_{,A} ## leave S (action) unchanced. For this reason only physical perturbations are considered. So it can be written as: $$\delta x^{\mu}=\delta x^R n^{\mu}_R$$
4. Intoduced the second fundamental form ##\Omega_{R,AB} ## and the normal fundamental form ##\mu_{R,SA} ## which defined for a given configuration of the strings world-sheet:
$$ \Omega_{R,AB}=g_{\mu \nu}n^{\mu}_R x^{\rho}_{,A } \nabla_{\rho}x^{\nu}_{,B}$$
$$ \mu_{RS,A}=g_{\mu \nu}n^{\mu}_R x^{\rho}_{,A } \nabla_{\rho}n^{\nu}_S$$
where ##\nabla_{\rho}## is the spacetime covariant derivative.
5. $$ \delta G_{AB}= -2\Omega_{R,AB} \delta x^R $$
And finally the second variation is found in the following form:
$$ \delta^2 S=\int d\tau d\sigma \sqrt {-h} ( \delta h_{AB}[2G^{BC} h^{AD} - \frac {1} {2} h^{AD}h^{BC} G^E_{ E} - \frac {1} {2} h^{AB} G^{CD} ]\delta h_{CD} + 4\delta h_{AB} h^{AC}h^{BD} \Omega_{R,CD} \delta x^R \\ -2\delta x^R[\delta_{RS} \Box - h^{AB} g_{\mu \nu} (x^{\rho}_{,A} \nabla_{\rho} n^{\mu}_R)(x^{\sigma}_{,B}\nabla_{\sigma}n^{\nu}_S)-2h^{AB} \mu_{RS,A} \partial_B - h^{AB}x^{\mu}_{,A}x^{\nu}_{,B} R_{\mu \rho \sigma \nu} n^{\rho}_R n^{\sigma}_S]\delta x^S)$$
where ## R_{\mu \rho \sigma \nu}## is the Riemann curvature tensor in the spacetime in which the string is embedded.
My question is how the second variation is obtained? I have calculated several times. But no there is no the same result.
An arbitrary string configuration is in 4-dimensional gravitational background. Starting point is Polyakov action
$$ S = \int d \tau d\sigma \sqrt {-h} h^{AB} G_{AB}$$.
Here is ##h_{AB}## is internal metric with determinant ##h##. ##G_{AB}## is the induced metric on the world-sheet:
$$G_{AB}=g_{ \mu \nu} \frac {\partial x^{\mu}} {\partial \xi^A} \frac {\partial x^{\nu}} {\partial \xi^B} = g_{ \mu \nu} x^{\mu}_{,A} x^{\nu}_{,B}$$
Internal metric is two dimensional metric of world-sheet coordinates ##\xi^A## (A=0,1), ## (\xi^0, \xi^1)=(\tau, \sigma) ##. ##x^{\mu} (\mu=0,1,2,3) ## are the spacetime coordinates.
Variations of the action with respect to ## \delta x^{\mu} ## and ## \delta h_{AB} ## is
$$ \delta S = \int d \tau d \sigma \sqrt {-h} \left( [ \frac 1 2 h^{AB} G^C_{ C} - G^{AB}] \delta h_{AB} - 2g_{\mu \nu} [\Box x^{\nu} + h^{AB} \Gamma^{\nu}_{ \rho \sigma} x^{\rho}_{,A}x^{\sigma}_{,B}] \delta x^{\mu} \right)$$
Here ##G^C_{ C}=h^{BC}G_{BC}## is the trace of induced metric on the world-sheet and ##\Box ## is the d'Alambertian:
$$ \Box = \frac {1} {\sqrt{-h}} \partial_A(\sqrt{-h} h^{AB} \partial_B). $$
From ## \delta S## is derived the equations of motion as usual
$$\frac 1 2 h^{AB} G^C_{ C} - G^{AB}=0$$
$$\Box x^{\nu} + h^{AB} \Gamma^{\nu}_{ \rho \sigma} x^{\rho}_{,A}x^{\sigma}_{,B}=0. $$
I got the first variations exactly as in paper (but with plus sign between the variations ## \delta h_{AB}## and ## \delta x^{\mu}##). But, equations of perturbations are obtained from the second variations of action and the problem starts here.
One more variation from the first variation is taken. And before doing this is done the following assumptions.
1. ## x^{\mu}=x^{\mu}(\xi^A)## is the solution of the equations of motion.
2. Introduced 2 vectors ##n^{\mu}_R (R=2,3)## normal to the surface of the string world-sheet:
$$ g_{\mu \nu} n^{\mu}_R n^{\nu}_S=\delta_{RS}, \\ g_{\mu \nu}x^{\mu}_{,A} n^{\nu}_R =0. $$
3. The general perturbation ## \delta x^{\mu}## can be composed as: $$\delta x^{\mu}=\delta x^R n^{\mu}_R+\delta x^A x^{\mu}_{,A} $$
The variations ##\delta x^A x^{\mu}_{,A} ## leave S (action) unchanced. For this reason only physical perturbations are considered. So it can be written as: $$\delta x^{\mu}=\delta x^R n^{\mu}_R$$
4. Intoduced the second fundamental form ##\Omega_{R,AB} ## and the normal fundamental form ##\mu_{R,SA} ## which defined for a given configuration of the strings world-sheet:
$$ \Omega_{R,AB}=g_{\mu \nu}n^{\mu}_R x^{\rho}_{,A } \nabla_{\rho}x^{\nu}_{,B}$$
$$ \mu_{RS,A}=g_{\mu \nu}n^{\mu}_R x^{\rho}_{,A } \nabla_{\rho}n^{\nu}_S$$
where ##\nabla_{\rho}## is the spacetime covariant derivative.
5. $$ \delta G_{AB}= -2\Omega_{R,AB} \delta x^R $$
And finally the second variation is found in the following form:
$$ \delta^2 S=\int d\tau d\sigma \sqrt {-h} ( \delta h_{AB}[2G^{BC} h^{AD} - \frac {1} {2} h^{AD}h^{BC} G^E_{ E} - \frac {1} {2} h^{AB} G^{CD} ]\delta h_{CD} + 4\delta h_{AB} h^{AC}h^{BD} \Omega_{R,CD} \delta x^R \\ -2\delta x^R[\delta_{RS} \Box - h^{AB} g_{\mu \nu} (x^{\rho}_{,A} \nabla_{\rho} n^{\mu}_R)(x^{\sigma}_{,B}\nabla_{\sigma}n^{\nu}_S)-2h^{AB} \mu_{RS,A} \partial_B - h^{AB}x^{\mu}_{,A}x^{\nu}_{,B} R_{\mu \rho \sigma \nu} n^{\rho}_R n^{\sigma}_S]\delta x^S)$$
where ## R_{\mu \rho \sigma \nu}## is the Riemann curvature tensor in the spacetime in which the string is embedded.
My question is how the second variation is obtained? I have calculated several times. But no there is no the same result.
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