Goldstein, who was born at Gleiwitz (now Gliwice in Poland), studied for a year at the University of Breslau (1869–70) then worked with Hermann von Helmholtz at the University of Berlin. He was appointed physicist at the Berlin Observatory in 1878, took his doctorate in 1881, and later established his own laboratory. In 1927 he became head of the astrophysical section of the Potsdam Observatory.

Goldstein's best-remembered scientific work is his studies of electrical discharges in gases at low pressures. He gave the name ‘cathode rays’ to the invisible emanations coming from the cathode of an evacuated discharge tube, showed that the rays could cast sharp shadows, and demonstrated that they were emitted perpendicular to the cathode surface. He later showed that they could be deflected by magnetic fields.