In the film, Man of Steel, we are told that Zod and his minions are being sent to the Phantom Zone for 300 cycles.
Is the Zone just a black hole?
Is the actual prison the pods that Zod's and his crew were stored in?
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The Phantom Zone is a prison dimension used by Kryptonians. It was originally discovered by Jor-El and used by the Kryptonian Council as a humane prison. Although the zone is a barren wasteland, people trapped in the zone can never get old or die, and can see outside of the zone and watch the events of the outside world.
The zone first appeared in the silver-age comics of the 1960s. It was used frequently in the Superman comics before the continuity was rebooted in the 1980s, and has appeared occasionally since. It has also appeared in other adaptations of Superman.
The Phantom Zone is an interdimensional realm outside the normal space/time continuum. It is a barren and insubstantial null area absent of any physical material. There is only one native denizen to the Phantom Zone, the enigmatic and powerful entity known as the Aethyr.
But it seems that like with any other comic book, thing, the phantom zone properties are altered to best served the current story.
In the post-Crisis DC Universe, the Phantom Zone first appears after Superman returns from space with a Kryptonian artifact called the Eradicator. This device, created by his Kryptonian ancestor Kem-L, attempts to recreate Krypton on Earth, building the Fortress of Solitude; the extradimensional space in which the Eradicator finds the Kryptonian materials necessary is called the Phantom Zone. A Phantom Zone Projector is part of Superman's current Fortress. It has been used to access the Bottle City of Kandor and to trap villains such as the White Martians.
The Phantom Zone is in the movie a prison that's most likely in another dimension...
As the wikia states
In the 2013 reboot film Man of Steel, General Zod and his fellow rebels are sentenced to 300 cycles of somatic reconditioning in the Phantom Zone following their attempted coup d'état against the Kryptonian government. Upon sentencing, Zod and his co-conspirators are infused within a gelatinous substance, encased in a crystalline material and are subsequently loaded into a Kryptonian ship. The ship then launches into orbit around Krypton where three smaller vessels establish a window into the Phantom Zone into which the ship enters. A short time later, the destruction of Krypton triggers the release of the prisoners. Later in the film, it is revealed that the vessels Zod and his army are using possess a "Phantom Drive", a collision from a smaller ship (piloted by Col Hardy of the U.S. Military with Kal-El's rocket and operated by Emil Hamilton and Lois Lane) with a similar drive causes a cataclysmic reaction that creates a small singularity, returning the ship and its occupants to the Phantom Zone, along with the Military plane, Hardy, and Hamilton. The mere fact that a couple of Kryptonian ships are needed to create a window into this zone shows that it's not a naturally occurring prison. As previously stated we can assume that it's most likely much like the Phantom Zones in other media.