- #1

Brinx

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For a course on tensor analysis, we were asked to perform some calculations regarding a 2D space embedded in 'regular' Euclidean 3D space. I've gotten some results, but I've hit a snag...

The following functions relate the coordinates in the embedding (3D, Euclidean) space to the coordinates in the embedded (2D) space (they describe the 'shape' of the embedded space):

[tex]X^1 = r \sin{\phi}[/tex] (1)

[tex]X^2 = r\cos{\phi}[/tex] (2)

[tex]X^3 = \log{r}[/tex] (3)

where the [itex]X^n[/itex] coordinates represent the coordinates in the embedding space, and [itex]r[/itex] and [itex]\phi[/itex] represent the radial and angular coordinates on the (apparently funnel-shaped) embedded space.

From this information, the following relations can be derived after some calculation (L is a constant):

[tex]\frac{d\phi}{ds} = \frac{L}{r^2}[/tex] (4)

and

[tex](1+r^2)\left(\frac{dr}{ds}\right)^2 = r^2 - L^2[/tex]. (5)

I was asked to derive these results, which I did (intermediate results given below) - starting from calculating the metric in the embedded space, then calculating Christoffel symbols of 1st and 2nd kinds, and using them calculating the shape of the geodesic equations and the kinematic condition.

Now, the question that's giving me some trouble is:

Induced metric from embedding:

[tex]g_{\mu\nu}(x)=G_{AB}(X(x))\frac{\partial F^A(x)}{\partial x^{\mu}}\frac{\partial F^B(x)}{\partial x^{\nu}}[/tex]

Relation between contra- and covariant metric components:

[tex]g^{\mu\nu}g_{\nu\alpha}=\delta^{\mu}_{\alpha}[/tex]

Christoffel symbol of the first kind:

[tex]\Gamma_{\mu\nu\alpha}=\frac{1}{2}(g_{\mu\nu,\alpha} + g_{\mu\alpha,\nu}-g_{\nu\alpha,\mu})[/tex]

Christoffel symbol of the second kind:

[tex]\Gamma^{\mu}_{\nu\alpha}=g^{\mu\beta}\Gamma_{\beta\nu\alpha}[/tex]

Geodesic equation (u is position differentiated w.r.t. arc length s, i.e. 'proper motion'):

[tex]\dot{u}^{\mu} + \Gamma^{\mu}_{\alpha\beta}u^{\alpha}u^{\beta}=0[/tex]

Kinematic condition:

[tex]u_{\mu}u^{\mu}=1[/tex]

Intermediate results, using the standard formulae:

The components of the (covariant) metric for the embedded space:

[tex]g_{11}=1 + \frac{1}{r^2}[/tex]

[tex]g_{22}=r^2[/tex]

[tex]g_{12} = g_{21} = 0[/tex]

Contravariant metric components:

[tex]g^{11} = \frac{r^2}{r^2 + 1}[/tex]

[tex]g^{22}=\frac{1}{r^2}[/tex]

[tex]g^{12}=g^{21}=0[/tex]

Christoffel symbols of the first kind:

[tex]\Gamma_{111}=\frac{-1}{r^3}[/tex]

[tex]\Gamma_{122}=-r[/tex]

[tex]\Gamma_{212} = \Gamma_{221} = r[/tex]

(other Christoffel symbols of the 1st kind are 0)

Christoffel symbols of the second kind:

[tex]\Gamma^{1}_{11}=\frac{-1}{r^3 + r}[/tex]

[tex]\Gamma^{1}_{22}=\frac{-r^3}{r^2+1}[/tex]

[tex]\Gamma^{2}_{12} = \Gamma^{2}_{21}=\frac{1}{r}[/tex]

(other Christoffel symbols of the 2nd kind are 0)

The two geodesic equations (dots denote differentiation with respect to s, the arc length):

[tex]\ddot{r} - \frac{1}{r^3+r}\dot{r}^2 - \frac{r^3}{r^2+1}\dot{\phi}^2 = 0[/tex]

[tex]\ddot{\phi} + \frac{2}{r}\dot{r}\dot{\phi}=0[/tex]

The kinematic condition:

[tex](1+\frac{1}{r^2})\dot{r}^2 + r^2\dot{\phi}^2=1[/tex]

From the second geodesic equation, equation (4) can be obtained by multiplying with r^2 and treating the result as a total derivative:

[tex]r^2\ddot{\phi} + 2r\dot{r}\dot{\phi}=\frac{d}{ds}(r^2\dot{\phi})=\frac{d}{ds}(L)=0[/tex]

Equation (5) follows from the kinematic condition (introducing L as described above and shuffling some terms around).

Now, to get back to the original question: for r(s) to be a constant means that all derivatives of r with respect to s must vanish. Looking at the first geodesic equation, the only way in which that can happen is when the derivative of [itex]\phi[/itex] with respect to s is also zero. And that means the only possible solution is a stationary point (i.e. r and [itex]\phi[/itex] both are constants). But that is in direct conflict with the kinematic condition, which would in that case state 0 + 0 = 1. So it seems there is no solution to the posed question!

Am I making a mistake somewhere?

## Homework Statement

The following functions relate the coordinates in the embedding (3D, Euclidean) space to the coordinates in the embedded (2D) space (they describe the 'shape' of the embedded space):

[tex]X^1 = r \sin{\phi}[/tex] (1)

[tex]X^2 = r\cos{\phi}[/tex] (2)

[tex]X^3 = \log{r}[/tex] (3)

where the [itex]X^n[/itex] coordinates represent the coordinates in the embedding space, and [itex]r[/itex] and [itex]\phi[/itex] represent the radial and angular coordinates on the (apparently funnel-shaped) embedded space.

From this information, the following relations can be derived after some calculation (L is a constant):

[tex]\frac{d\phi}{ds} = \frac{L}{r^2}[/tex] (4)

and

[tex](1+r^2)\left(\frac{dr}{ds}\right)^2 = r^2 - L^2[/tex]. (5)

I was asked to derive these results, which I did (intermediate results given below) - starting from calculating the metric in the embedded space, then calculating Christoffel symbols of 1st and 2nd kinds, and using them calculating the shape of the geodesic equations and the kinematic condition.

Now, the question that's giving me some trouble is:

*- Show that, for a given L > 0, there exists a geodesic along which r(s) is a constant, and calculate this constant.*## Homework Equations

Induced metric from embedding:

[tex]g_{\mu\nu}(x)=G_{AB}(X(x))\frac{\partial F^A(x)}{\partial x^{\mu}}\frac{\partial F^B(x)}{\partial x^{\nu}}[/tex]

Relation between contra- and covariant metric components:

[tex]g^{\mu\nu}g_{\nu\alpha}=\delta^{\mu}_{\alpha}[/tex]

Christoffel symbol of the first kind:

[tex]\Gamma_{\mu\nu\alpha}=\frac{1}{2}(g_{\mu\nu,\alpha} + g_{\mu\alpha,\nu}-g_{\nu\alpha,\mu})[/tex]

Christoffel symbol of the second kind:

[tex]\Gamma^{\mu}_{\nu\alpha}=g^{\mu\beta}\Gamma_{\beta\nu\alpha}[/tex]

Geodesic equation (u is position differentiated w.r.t. arc length s, i.e. 'proper motion'):

[tex]\dot{u}^{\mu} + \Gamma^{\mu}_{\alpha\beta}u^{\alpha}u^{\beta}=0[/tex]

Kinematic condition:

[tex]u_{\mu}u^{\mu}=1[/tex]

## The Attempt at a Solution

Intermediate results, using the standard formulae:

The components of the (covariant) metric for the embedded space:

[tex]g_{11}=1 + \frac{1}{r^2}[/tex]

[tex]g_{22}=r^2[/tex]

[tex]g_{12} = g_{21} = 0[/tex]

Contravariant metric components:

[tex]g^{11} = \frac{r^2}{r^2 + 1}[/tex]

[tex]g^{22}=\frac{1}{r^2}[/tex]

[tex]g^{12}=g^{21}=0[/tex]

Christoffel symbols of the first kind:

[tex]\Gamma_{111}=\frac{-1}{r^3}[/tex]

[tex]\Gamma_{122}=-r[/tex]

[tex]\Gamma_{212} = \Gamma_{221} = r[/tex]

(other Christoffel symbols of the 1st kind are 0)

Christoffel symbols of the second kind:

[tex]\Gamma^{1}_{11}=\frac{-1}{r^3 + r}[/tex]

[tex]\Gamma^{1}_{22}=\frac{-r^3}{r^2+1}[/tex]

[tex]\Gamma^{2}_{12} = \Gamma^{2}_{21}=\frac{1}{r}[/tex]

(other Christoffel symbols of the 2nd kind are 0)

The two geodesic equations (dots denote differentiation with respect to s, the arc length):

[tex]\ddot{r} - \frac{1}{r^3+r}\dot{r}^2 - \frac{r^3}{r^2+1}\dot{\phi}^2 = 0[/tex]

[tex]\ddot{\phi} + \frac{2}{r}\dot{r}\dot{\phi}=0[/tex]

The kinematic condition:

[tex](1+\frac{1}{r^2})\dot{r}^2 + r^2\dot{\phi}^2=1[/tex]

From the second geodesic equation, equation (4) can be obtained by multiplying with r^2 and treating the result as a total derivative:

[tex]r^2\ddot{\phi} + 2r\dot{r}\dot{\phi}=\frac{d}{ds}(r^2\dot{\phi})=\frac{d}{ds}(L)=0[/tex]

Equation (5) follows from the kinematic condition (introducing L as described above and shuffling some terms around).

Now, to get back to the original question: for r(s) to be a constant means that all derivatives of r with respect to s must vanish. Looking at the first geodesic equation, the only way in which that can happen is when the derivative of [itex]\phi[/itex] with respect to s is also zero. And that means the only possible solution is a stationary point (i.e. r and [itex]\phi[/itex] both are constants). But that is in direct conflict with the kinematic condition, which would in that case state 0 + 0 = 1. So it seems there is no solution to the posed question!

Am I making a mistake somewhere?

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