How do I differentiate this Scalar Field?

[unscientific]

Homework Statement

(a) Find the christoffel symbols (Done).
(b) Show that $\phi$ is a solution and find the relation between A and B.[/B]

Homework Equations

The Attempt at a Solution

Part(b)
$$\nabla_\mu \nabla^\mu \phi = 0$$
I suppose for a scalar field, this is simply the normal derivative:
$$\frac{\partial^2 \phi}{\partial \eta^2} + \frac{\partial^2 \phi}{\partial r^2} = 0$$

Starting with the $\eta$ component:
$$\frac{\partial \phi}{\partial \eta} = exp() \left[ B + (A+B\eta)(-ic|k|) \right]$$
$$\frac{\partial^2 \phi}{\partial \eta^2} = exp() \left[ -2ic|k|B - |k|^2c^2(A+B\eta) \right]$$

Now for the $r$ component:
$$\frac{\partial \phi}{\partial r} = exp() \left[ A+B\eta \right](\vec k \cdot \hat r)$$
$$\frac{\partial^2 \phi}{\partial r^2} = exp() \left[ -(A+B\eta)(\vec k \cdot \hat r)^2 \right]$$

Equating both real parts doesn't work; It gives $(\vec k \cdot \hat r)^2 = |k|^2 c^2$..

 

[strangerep]

What is $\nabla_\mu$ here? I would have thought they'd be covariant derivatives, not partial derivatives. (?)

Even though a covariant derivative of a scalar field is just the partial derivative, the 2nd covariant derivative is acting on a vector...

 

[Orodruin]

strangerep is correct, the second covariant derivative is not acting on a scalar, but on a covariant vector. I suggest rewriting it as
$$
\nabla^\mu \nabla_\mu \phi = g^{\mu\nu} \nabla_\mu \nabla_\nu \phi
$$
and apply the rules for covariant derivatives from there.

 

[unscientific]

strangerep is correct, the second covariant derivative is not acting on a scalar, but on a covariant vector. I suggest rewriting it as
$$
\nabla^\mu \nabla_\mu \phi = g^{\mu\nu} \nabla_\mu \nabla_\nu \phi
$$
and apply the rules for covariant derivatives from there.

I know that for a vector field, $\nabla_\alpha V^\mu = \partial_\alpha V^\mu + \Gamma^\mu_{\alpha \gamma} V^\gamma$. Does this apply to scalar fields as well? Or in the case of a scalar field, does it simply reduce to a normal derivative?

I have also found that
$$\nabla_\alpha \nabla_\beta V^\mu = \partial_\alpha \partial_\beta V^\mu + \left(\partial_\alpha \Gamma^\mu_{\beta \nu} \right)V^\nu + \Gamma^\mu_{\beta \nu} \left( \partial_\alpha V^\nu \right) + \Gamma^\mu_{\alpha \nu} \left[ \partial_\beta V^\nu + \Gamma^\nu_{\beta \epsilon} V^\epsilon \right] + \Gamma^\nu_{\alpha \beta} \left[ \partial_\nu V^mu + \Gamma^\mu_{\nu \epsilon} V^\epsilon \right]$$

 

[Orodruin]

I know that for a vector field, $\nabla_\alpha V^\mu = \partial_\alpha V^\mu + \Gamma^\mu_{\alpha \gamma} V^\gamma$. Does this apply to scalar fields as well? Or in the case of a scalar field, does it simply reduce to a normal derivative?

No, for scalar fields you have $\nabla_\mu \phi = \partial_\mu \phi$. However, $\nabla_\mu \phi$ is a covector field and as such its covariant derivative $\nabla_\nu\nabla_\mu \phi$ will involve a Christoffel symbol.