Factual error: One of the programs on the new television set is Animal, Vegetable, Mineral? In 1953, this quiz show was broadcast on Thursdays. But, it's the eve of the coronation, which makes it Monday the 1st of June. Animal, Vegetable, Mineral? wasn't on that evening.
Factual error: The Doctor (and the staff on the station) refer to the idea of a planet being in orbit around a black hole as 'impossible'. It is not. Gravity (and physics in general) works perfectly well outside of the event horizon. For all practical purposes - regarding orbiting around it - the black hole might as well be any other object, as long as it has an equal mass. (00:07:15 - 00:08:30)
Continuity mistake: When the Doctor and Ida descend into the core of Krop Tor, the screen charting their progress shows 39% Oxygen. After cutting away to an overhead shot of the two, the camera cuts back to the same descent display, only now the oxygen percentage (which has been descending constantly all this time) is at 42%. (00:35:25)
Factual error: Ida, the science officer, says that the planet is in geostationary orbit around the black hole. However, the word "geostationary" applies very specifically to objects orbiting the planet Earth. Since a black hole is a type of star, this planet's orbit could be described as astrostationary, or even just stationary, but definitely not as geostationary. A science expert on an interstellar mission wouldn't make this mistake, and she wasn't dumbing things down, either, since "geostationary orbit" is already a pretty obscure topic for people unfamiliar with space technology. (00:07:40)
Plot hole: Those people created for medical experiments cannot run. In fact they can barely walk, so how come they can climb nearly as fast as Rose? What's more, when you see them climbing, they seem to have little ability to coordinate hand over hand. Even with the delay brought on by the cat-woman-thing, they should not have been able to keep up with her.
Factual error: When the Doctor regrows his hand during the swordfight, he says that he can do this because he's still within the first 15 hours of his regeneration cycle, and there's still residual energy in his body. However: In both this scene, and the episode's opening scene, when the TARDIS arrives in London, it is full daylight. The episode takes place on Christmas Eve and Christmas Day, 3-4 days after the Winter Solstice. London, at 51°N latitude, has a night longer than 15 hours at that time of year, meaning the Doctor should have been out of luck.
Continuity mistake: When Rose presents her mother with the Bazoolium, she's holding it in her left hand. When the view switches to look at Jackie, it shifts to Rose's right hand and then immediately back to her left when the view restores to the original vantage point. (00:02:45)
Revealing mistake: During the scene in which the Doctor is climbing the transmitter, there is a shot in which he is viewed from above and is then struck by lightning. In this shot, the Doctor's foot disappears briefly as it leaves the green screen on the ground under the actor. (00:38:25)
Factual error: The graphics depicting the black hole show matter falling directly towards the centre of it. This would only happen if all the matter happened to be heading in that direction already. In reality, matter approaching the black hole at any other angle would swirl around the event horizon, like water going down a plughole.
Revealing mistake: When the Doctor and Rose are talking at the bay, the Doctor's hair is blowing in the wind, but it's not when he is shown in the TARDIS. Considering the image of him on the beach is a projection of him in the TARDIS, the two versions should match. The actor was obviously on the beach recording his lines with Rose and back on a set doing the inside scenes.
Continuity mistake: When Mickey enters one of the school's computer rooms, dozens of pupils are busy typing, and every one of them is also wearing headphones. However, once Mickey unplugs the main connection for the computers, and shouts for the students to leave, when the students begin to get up, their headphones have instantly removed themselves from their heads and been placed neatly on the desks next to the computer keyboards. (00:36:25)