  We develop a first-principles scaling theory of the spreading of
three-dimensional (3D) magnetic reconnection of finite extent in the out of
plane direction. This theory addresses systems with or without an out of plane
(guide) magnetic field, and with or without Hall physics. The theory reproduces
known spreading speeds and directions with and without guide fields, unifying
previous knowledge in a single theory. New results include: (1) Reconnection
spreads in a particular direction if an x-line is induced at the interface
between reconnecting and non-reconnecting regions, which is controlled by the
out of plane gradient of the electric field in the outflow direction. (2) The
spreading mechanism for anti-parallel collisionless reconnection is convection,
as is known, but for guide field reconnection it is magnetic field bending. We
confirm the theory using 3D two-fluid and resistive-magnetohydrodynamics
simulations. (3) The theory explains why anti-parallel reconnection in
resistive-magnetohydrodynamics does not spread. (4) The simulation domain
aspect ratio, associated with the free magnetic energy, influences whether
reconnection spreads or convects with a fixed x-line length. (5) We perform a
simulation initiating anti-parallel collisionless reconnection with a pressure
pulse instead of a magnetic perturbation, finding spreading is unchanged rather
than spreading at the magnetosonic speed as previously suggested. The results
provide a theoretical framework for understanding spreading beyond systems
studied here, and are important for applications including two-ribbon solar
flares and reconnection in Earth's magnetosphere.
