I will try a different approach – I suppose it is good to consider different angles on how this collision can happen...
Speaking on approach, if this is a collision... from where does the steel come into the mountain? I need to know that in order to know where the energy goes to.
Let us say I have a huge lamp in my backyard, and this lamp is designed to work in outer space, and has a built-in battery. Then I turn up the lamp, afterwards, I teleport it up (away from earth)... say 80% the distance to the moon. And I look up... I will not see the light from the lamp until about a second after I send it, because that is how much time light takes to go from the lamp to me.
If we consider a space-time diagram (where space is in the horizontal and time in the vertical), first the lamp is stationary, meaning that its world in moves only in time (it is ontop of the vertical axis), about a second per second. At some point I turn it on, we can represent all the places that the light of the lamp can reach as a cone that opens up. When I teleport the lamp, the world line of the lamp moved horizontally in such way that I was out of the light cone of the lamp... and as time passes, one second later (up in the diagram), I get back in the light cone (the light cone crosses the vertical axis).
Note: interestingly, the light cone from before the movement and after would at some point intersect. Meaning that there could be observers that can see the light from the lamp before it being teleport and after at the same time.
Now, what force do I need to apply to the teleport object such that it moves in this unusual way? Remember that for everything as we know it, time moves inexorably forward. We do not know of any force that when hitting an object, it would make time stop for that object (as in: if you were hit, you would see the world held in place).
We could try to calculate what that force would be, and we would come up with infinite... and it would make the object move faster than the speed of light, and it would technically be traveling to the past.
Note: In fact, if I send a probe to meet with the lamp at its destination, such that it arrives at the same time at the lamp gets there... when we do the Lorentz transformation for the probe, we would find that the lamp was send from the future according to the frame of reference of the probe.
Thus, we must conclude that we must hit the object with a force that moves from the future to the past.
If we consider force as a vector that has magnitudes in each dimension, I am proposing that all known physics is dealing with force vector that have 0 in the time dimension, and we need force vectors that have a negative value there (assuming positive means towards the future).
Now we know that we are talking about a collision from the future to the past. That’s the angle.
Let us say, we can push the object to the past. Why would it stop going to the past? Well, it could be because it collides with something. Or perhaps there is some form of drag (after all, there is something pushing everything inexorably to the future).
F = m*a; We are – of course – describing a movement, and assuming the mass does not change. Thus, hmm… we would have an acceleration vector with a negative value on the time coordinate.
Considering that acceleration is displacement (distance) over time squared, we are talking about an acceleration that will make the object perceive time slow down and then go backwards.
However, remember that we are not moving the object to the same place! We are sending it to a different location. That means that acceleration also must have space components.
The way I imagine (i.e. what I am about to say lacks any rigor): We might see the object start moving in the direction we teleport it, stretch (I'd blame Lorentz contraction), then disappear.
How can we do that push in the first place? There must be an equal an opposite reaction. And, no, that does not mean solving the teleport by exchange (a solution I am personally fond of, but not allowed by OP). We must be pushing something to the future with the same force. That is, the teleport machine has a form of recoil. Perhaps we need to send a small object away to the distant future; as consequence of the teleport of something heavy, a short distance to the short past.
Also, when the collision happens, it could push objects to the past and bounce to the future. Weird stuff. However, no, I am not going with that... instead...
Feynman diagrams has the oddity that when flipped in time they remain valid. Thus, they serve as a useful model for interactions with particles moving backward in time.
What we find is that – and I oversimplifying here – a particle moving backwards in time, is its own antiparticle moving forwards in time. That means that an object moving backwards in time, will be – for the rest of us – made of antimatter.
So, about your steel sphere to appear instantaneously in solid rock… yeah, boom. It annihilates with the solid rock.
The following computations are done with Wolfram|Alpha
Well, you told me that it is a sphere that measure 1m on diameter. That is 0.523599 cubic meters of steel. Going with 7900 kg/m^3 for the steel density, we have 4136.43 kg. Then, 2650 kg/m^3 for nondescript rock, we have another 1387.54 kg; for a total of 5523.97 kg.
5523.97 kg gives us 4.9647×10^20 J by good old Einstein’s equation. That is an explosion equivalent to 1.1866×10^11 tons of TNT (118.7 gigatons of TNT).
I struggle to express this…
The following information is from Wikipedia
The Tunguska event is estimated to be 3×10^7 tons of TNT. And Tsar Bomb is only 5×10^7 tons of TNT. Those are four orders of magnitude smaller. Thus, we are looking for something more devastating.
The dinosaur killer is 1.92×10^14 tons of TNT. Three orders of magnitude bigger. So, not that devastating.
The following is from Science Magazine
I found an article about a crater found under Greenland that is said to have required an impact of 3×10^21 J or 7.17×10^11 tons of TNT. This is the right order of magnitude. The article says:
The impact would initially produce a bowl-shaped cavity ~20 km in diameter and ~7 km deep, which would quickly collapse (within ~1 min) to form a complex crater more than 31 km in diameter and ~800 m deep with a central uplift. This impact scenario would have melted and vaporized up to ~20 km^3 of target rock, approximately half of which would have remained within the crater, forming a melt sheet up to ~50 m deep.
From the point of view of the object... does its own time continues on during the teleport? Perhaps it does not, it also perceives the teleport as instantaneous.
Where is the object in the instant of the teleport, according to the object? My intuition is that it must be in all places in the path from its origin to its destination. If you think of it that way, you would see that it must hit with the near past of something close to your lab before it can hit the target mountain. And the collision does not look good for your lab.
Perhaps it is preferable to think that the teleport works by moving the object through "hyperspace" and thus avoiding any undesired collision. Moving into hyperspace will also be moving in an additional dimension. In that case... well, whatever it hits at its destination is pushed into hyperspace.
How does a particle get from one place to another? Currently, as far as quantum theory goes, we must consider as if the particle takes all the infinite possible paths from the origin to the destination. That includes paths that go back in time, and paths that go through barriers of other objects. However, the effects of some of these paths cancel each other out, and some paths have a higher influence than others according to the wave function.
Thus, in order to compute the future state of a quantum system, we start by the more relevant paths and move to the less relevant ones, and by doing so we approximate the solution.
Now, at the moment of interaction, the quantum state of the particle must be solved, and it will be at a given place... and to resolve what happens as a result of the interation, we do what I described above. In fact, this is the use of Faymann diagrams, they are used to categorize the possible things that could happen in such way that we can organize them and start computing.
Then the problem with extending quantum tunelling to massive objects is that massive objects are formed by particles that are constantly interacting with each other, keeping the system coherent. That is also the solution to Schrödinger's cat. Or if you prefer Einstein's version that has a bomb that explode or not, that is why you do not see the box explode and not explode at the same time.
However, that is not to say that an object cannot spontaneusly teleport under this framework. It would just be absurdly unlikely that all of its particle undergo decoherence simultanously, tunnel to another place and become coherent in the same arrangement.
We do not really know enough of the why quantum physics works. For instance, it is possible to think about the wave function of complex objects, or even the wave function of the whole universe. And, well... is the universe interacting with something else to keep it coherent?
So, let us go with OP and say that this is not only possible but that we can create a machine to make it happen. Well, in this case, we have an elegant coup-out: if a particle cannot occupy a position because there is another particle there (Pauli exclusion principle), then the probability of the object appearing there is zero, thus, it will not happen, instead the steel sphere will appear somewhere else.
Addendum: As kingledion, Mark Olson and Carl Sagan point out "Matter is composed chiefly of nothing". There is a lot of empty space in the atoms. However, if the teleport respect pauli exclusion principle (and should it not?) then the particles of the steel sphere will appear is such way that no nuclear reaction happens, only chemical. Thus, you could still have the steel sphere embeded in the rock, and it would be made of... hmm... a different steel, due to the rock chemistry.
Addendum: Don't Panic
Assuming you can build a machine that can cause highly improvable events such as the spontaneous and instantaneous displacement of an object, as mentioned above. Let us call it Teleport-MK1.
The Teleport-MK1 would still not be able to break the exclusion principle. That does not mean it is useless… for example, you could use it leap people's underwear a meter※ to the left as a party trick.
※: Because it is safer that just moving it one foot, and metric is better™.
However, a more interesting use is to have it arrange matter in any desired configuration, as long as it is possible. This device could transform virtually any raw material into any physically possible object, including food (or very advanced substitutes), and other machines. Let us call this reconfiguration of the Teleport-MK1 the Replicator-MK1.
If the universe allows the creation of a machine capable causing impossible events, then you can use a Replicator-MK1 to create it. The new device would not only be able to teleport and manufacture, it would be able to do anything, even if it is infinitely improbable.
What happens when you use it to teleport a steel sphere inside a mountain? Whatever you want.
Note: Possible side effects of the use of the new device include time traveling bowls of petunias, use with care. Read Douglas Adams books on the subject for more information. Whatever happens, remember: Don't Panic.
When the steel spehre collides with the mountain, it will either:
- Swap places with the rock
- Cause a huge explosion
- Push the rock into hyperspace
- The steel sphere appears somewhere else (the universe forbids the collision)
- The steel sphere chemically merges with the rock
- Whatever you want
(this is not an exhaustive list of things that could happen, just the ones I mention in my answer)