Gravitational lensing and rotation curve

Based on the geodesic equation in a static spherically symmetric metric we discuss the rotation curve and gravitational lensing. The rotation curve determines one function in the metric without assuming Einstein’s equations. Then lensing is considered in the weak field approximation of general relativity. From the null geodesics we derive the lensing equation. The gravitational potential U(r) which determines the lensing is directly give by the rotation curve U(r) = −v ²(r). This allows to test general relativity on the scale of galaxies where dark matter is relevant.