A shaft (12) for cooling an electric machine (10) includes: an outer sub-shaft (32), which is designed for being rotationally fixed to a rotor (14) of the electric machine; an inner sub-shaft (30), which is rotationally fixed to the outer sub-shaft and is designed as an output shaft of the electric machine; an inflow (36) arranged in the radial direction between the outer sub-shaft and the inner sub-shaft, in order to supply cooling fluid (28) to the shaft; and an outflow (42) arranged in the radial direction between the outer sub-shaft and the inner sub-shaft, in order to discharge cooling fluid supplied to the shaft. The outer sub-shaft encloses a fluid chamber (40), which is arranged in the axial direction between the inflow and the outflow. A direction of flow of cooling fluid in the fluid chamber is established by a delivery direction of a fluid pump (24).