Material Sources for the Kerr Metric

The Kerr metric is a vacuum solution of the Einstein field equations which appears to be the field exterior to some axially symmetric, rotating body. A method based on Synge's $g$ method (guess $g_{\mu \nu }$ and calculate $T_{\mu \nu }$) is given for constructing interior solutions of the field equations which describe rotating, nonperfect-fluid bodies which might serve as sources of the Kerr metric. Finally, an argument is given which indicates that reasonable perfect-fluid—type solutions which might serve as sources of the Kerr metric may not exist.