The phrase in righteousness
refers to the justice that God promises to his people. Previous chapters of Jesus’ revelation to John had shown numerous instances where his people were oppressed, persecuted, even martyred for the faith (Revelation 6:9; Revelation 12:17; Revelation 13:15–16). As a righteous judge this rider vindicates his people, be it in judgment or war.
11 Then I saw heaven opened, and behold, a white horse! The one sitting on it is called Faithful and True, and in righteousness he judges and makes war.