❓What is the expression for the tidal tensor outside a spherical mass distribution?

The tidal tensor $T_j^i=-\frac{\partial^2 \phi}{\partial x^i \partial x^j}=\frac{\partial g^i}{\partial x^j}$ for the potential $\phi=-G M / r$ is calculated as follows (as shown in your document):

• Off-Diagonal Elements ( $i \neq j$ ):

  $T_j^i=\frac{\partial}{\partial x^j}\left(-\frac{G M x^i}{r^3}\right)=-G M x^i\left(-3 \frac{x^j}{r^5}\right)=\frac{3 G M x^i x^j}{r^5}$

• Diagonal Elements (i=j) :

  $T_i^i=\frac{\partial}{\partial x^i}\left(-\frac{G M x^i}{r^3}\right)=G M\left[\frac{3\left(x^i\right)^2-r^2}{r^5}\right]=G M\left[\frac{3\left(x^i\right)^2}{r^5}-\frac{1}{r^3}\right]$

The full tensor $T_{i j}$ can be compactly written as:

$$T_{i j}=G M\left(\frac{3 x^i x^j}{r^5}-\frac{\delta_{i j}}{r^3}\right)$$

where $\delta_{i j}$ is the Kronecker delta (equal to 1 if i=j and 0 if $i \neq j$ ).

The full tensor is defined by two terms: a stretching component and a compression/squeezing component.

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