- #1

user1139

- 72

- 8

- Homework Statement:
- Show that the d'Alembertian of the scalar $$1/R^2=0$$

- Relevant Equations:
- I am suppose to use this expression $$\partial_{\mu}R=\frac{\eta_{\mu\nu} x^{\nu}}{R}$$ to help show

Assuming Einstein summation convention, suppose $$R^2=\eta_{\mu\nu}x^{\mu}x^{\nu}$$

I was able to show that $$\partial_{\mu}R=\frac{\eta_{\mu\nu} x^{\nu}}{R}$$ by explicitly doing the covariant component of the four-gradient and using the kronecker tensor.

However, how do I use the equation expressed in the second paragraph to show that $$\eta^{\alpha \beta}\partial_{\alpha}\partial_{\beta}\frac{1}{R^2}=0$$? I tried $$R\rightarrow \frac{1}{R^2}$$ but I got expressions containing $$x^{\nu}x_{\alpha}$$ which will not give me 0 unless I assume $$x^{\nu}$$ and $$x_{\alpha}$$ are orthogonal which I think is wrong to do so.

I was able to show that $$\partial_{\mu}R=\frac{\eta_{\mu\nu} x^{\nu}}{R}$$ by explicitly doing the covariant component of the four-gradient and using the kronecker tensor.

However, how do I use the equation expressed in the second paragraph to show that $$\eta^{\alpha \beta}\partial_{\alpha}\partial_{\beta}\frac{1}{R^2}=0$$? I tried $$R\rightarrow \frac{1}{R^2}$$ but I got expressions containing $$x^{\nu}x_{\alpha}$$ which will not give me 0 unless I assume $$x^{\nu}$$ and $$x_{\alpha}$$ are orthogonal which I think is wrong to do so.