Not necessarily.
Airplanes can fly upside down (well, most of 'em…). Usually you roll them inverted (that’s movement about the long axis - lie in bed with your arms outstretched, then roll over from right to left and you’ll get the idea) rather than use pitch (that’s where the tail goes over the nose or vice versa, like if you did a somersault) but in theory both work. In fact, what mangeorge describes is a “loop”. Normally, you loop by going UP, because it’s not a good idea to get too close to the ground. The key thing in a standard loop is to maintain enough airspeed to carry you up and over to the point where gravity starts to pull you down, but late enough you’re coming down nose-first. (If gravity takes over before the top of the loop you come down tail-first. This is a Very Bad Thing is most cases)
Yes, in theory if you held the stick or yoke forward you the airplane would, essentially, perform a forward somersault and you would (eventually, in theory) find yourself upright again. Problem is, in practice, once you point the nose down gravity takes over and it is extremely unlikely you’d have either enough room to complete the manuver or enough power/control to come out of the resulting dive. But the manuver in and of itself is not likely to overstress the airframe and probably it’s been done by somebody somewhere succesfully (although probably not in an airliner). Excessive speed during the manuver - yes, that will cause damage. The question is, would the required airspeed to successfully complete such a loop in a Boeing 757 exceed the Vne? I don’t know. I don’t think anyone does. Such manuvers are not considered when designing passenger jets.
Airplanes are not quite like cars - when you turn the wheel of your car while driving and let go of the wheel, the wheel turns back to neutral and the car comes out of the turn. Airplane controls do tend to return to neutral… but when you turn the yoke to put it into a turn, while they DO tend to return to the neutral position when you let go of them the airplane tends to stay in a banked turn. They do try to design airplanes such as passenger jets so they tend to return to straight and level flight, but this take a lot longer to happen than it does for the steering wheel on your car to return your straight line of travel. Once you get that nose pointed down a significant degree it’s going to tend to stay in that position, even with the yoke moving to neutral on its own. At a certain point, gravity alone will be enough to continue the downward path on it’s own, or even increase the steepness of the dive. You don’t have to “point the nose straight down” to achieve this glorious state, either.
At this point we’re moving beyond my expertise - I’m not an authority on what is, essentially, stunt flying. But all aircraft obey the same laws of physics and all fixed wing aircraft have certain tendencies and limitations (rotorcraft have some different limitations and tendencies, as folks like Johnny L.A. can testify).
OK, let’s see if I can do this without the 40 minute lecture on aerodynamics (which even I would find boring…)
To a large degree, the difference between a “descent” and a “dive” in an airplane is a matter of degree only. I can tootle along straight and level at 110 mph, then adjust the controls so that I will descend at 110 mph even if I’m not touching the controls and even without benefit of autopilot or other mechanical aids. I may be descending at a 3 degree angle or a 10 degree angle or even a 30 degree angle which would be pretty steep - I got a “yikes!” out of a fellow pilot for doing that once without warning. Most people would call a 30 degree descent a “dive”. (Thing is, when I do it, it’s a controlled dive - I know what I’m doing, how I’m going to do it, and how I’m going to get out of it. It’s not an accident but something I plan and perform according to that plan) In other words, no, you don’t have to pull back to maintain the angle. In fact, if you point down steeply enough, you’ll have to pull back to keep the angle from getting worse. Yep, it’s different than motion and direction in ground vehicles. A lot of basic flight training is learning that difference so it gets down to your gut level and you can anticipate how the airplane will react.
Would the passengers know enough to delibrately put an airliner into this position? Depends on the passengers. The hijackers certainly had the knowledge, and, as I said, there was at least one other pilot among the passengers.
On the other hand, someone falling onto the controls during a fistfight could easily put the plane into this situation by accident, and then the question is - is there anyone aboard with the knowledge to get *out of *this situation?
Another factor is the autopilot. An autopilot keeps an airplane on course (in part) by counter-acting any forces like gusts, or perhaps an accidental bump on the controls. As a result, the airplane flies in a straight line. If the autopilot is set and you push on the yoke, the autopilot will push back. So, in our hypothetical cockpit fight, IF the autopilot was set while the bodies were flying about the interior, it would try to compensate for any jars and jolts. Up to a point.
You see, most autopilots will automatically disengage if a certain amount of movement is performed on the controls. For instance, if a pilot has to make a sudden manuver to avoid hitting another airplane, this will allow him to do that without having to first deactivate the autopilot - the assumption is that if the controls are moved that much, it’s time for the human to take over.
Here’s the rub - when the autopilot lets go, it really lets go. All the little compensating things it does to make up for, say, the yoke being more forward than it should be - stop. An autopilot unhitching itself due to sudden extreme inputs can result in a roller coaster-like ride if the pilot isn’t ready to take over immediately. (In some circumstances - such as heavy turbulence - pilots turn off autopilots to prevent these sorts of sudden ooopsies.)
So, let’s go back to Flight 93. The passengers storm the cockpit. A fight breaks out. At first the autopilot struggles to compenstate for the jars and bumps given the controls during the initial scuffle and keeps the airplane more or less on course. (that could account for some minor rolls and yaws and nosing up and down) However, the fighting intensifies and someone is thrown into the control yoke, or falls over it, shoving it forward far enough and quickly enough to deactivate the autopilot. The airplane noses over, there is perhaps a brief interlude of zero-g, then everyone and everything loose in the cockpit slams into the back wall of the cockpit as the acceleration builds up. This may result in the body draped across the yoke(s) falling away, relieving the pressure on the controls and allowing them to return to neutral, but by this point that does no good - gravity and the airflow around the jumbo is now helping to keep the nose down. The speed builds, passes Vne. At some point past that mark, pieces start to peel off the airplane - maybe nothing more than some skin panels, maybe an engine. Then comes the impact.
It would happen within a few seconds. From the viewpoint of anyone inside the airplane, assuming they were concious, they would be long seconds. Me, I’m not sure if I’d be frightened out of my wits or numb with terror. Hard to say - hope never to be in that position.
And it’s all speculation at this point - as I’ve said, we will never know all the details of what happened. There are a number of different ways this could have played out, including a delibrate nose-over suicide by the hijackers. We just don’t know for sure.