Consider the following integral $$ \int_{V_X(v)}I(Y)\,dY\tag1 $$ In light cone coordinates $x_\pm=x\mp vt$, where $t$ is time and $v\in\mathbb{R}^+$, $V_X(v)$ is the past light cone, given, by $$ V_X(v)=\{Y\text{ such that }x_+\leq y_+,y_-\leq x_-\} $$ Hence, integral $(1)$ can be given by $$ \int_{x_+\leq y_+,y_-\leq x_-} I(y_+,y_-)dy_+dy_-. $$ How can I obtain the time derivative $$ \frac{\partial}{\partial t} \int_{x_+\leq y_+,y_-\leq x_-} I(y_+,y_-)dy_+dy_- $$ Is there a closed formula?
2026-02-23 06:19:40.1771827580
How to differentiate a light-cone integral in relativity?
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