1. **State the problem:** We need to find the value of $y^2$ given the equation $$9x^2 - 9y^2 = 4.$$\n\n2. **Isolate the term involving $y^2$:** Subtract $9x^2$ from both sides to get:\n$$-9y^2 = 4 - 9x^2.$$\n\n3. **Divide both sides by -9 to solve for $y^2$:**\n$$y^2 = \frac{4 - 9x^2}{-9} = \frac{9x^2 - 4}{9}.$$\n\n4. **Simplify the expression:**\n$$y^2 = x^2 - \frac{4}{9}.$$\n\n**Final answer:** $$y^2 = x^2 - \frac{4}{9}.$$