In 2004 he was elected to the Australian Academy of Science, with citation: Professor Bartnik is renowned internationally for the application of geometric analysis to mathematical problems arising in Einstein's theory of general relativity.
His work is characterised by his ability to uncover new and anticipated phenomena in space-time geometry, often employing sophisticated tools from linear and nonlinear partial differential equations as well as elaborate numerical computations.
He has contributed greatly to our understanding of the properties of the Einstein equations and gravitation.His work with John McKinnon has been widely studied in the physics literature.
[1] In the mathematics literature, he was known for his work with Leon Simon on solving the Dirichlet problem for prescribed mean curvature, in the setting of spacelike hypersurfaces of Minkowski space.
He showed that it is geometrically defined by constructing "optimal" asymptotically flat coordinates, and gives an extension of Edward Witten's proof of the time-symmetric positive energy theorem to the higher-dimensional spin setting.