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I'm surprised I can't find a previous question on this.

We all know that being teleported instantly from one place to another has a big problem. If your co-ordinates are off, you might appear in the middle of a mountain or at best you appear in the correct place but intersecting with the air.

I think that the air problem is solved relatively easily.

When transporting inanimate objects, you could send them to a vacuum chamber thus avoiding collision problems. Similarly a human could be sent wearing a pressurised suit to avoid exposing them to a vacuum.

So here's the crunch.

What happens if the coordinates are wrong?

Question

A miscalculation or deliberate action causes a one metre diameter solid steel sphere to appear instantaneously in solid rock in the middle of Mont Blanc. Assume the teleportation is one-way so the rock stays where it is. (Thanks to kikirex for raising this)

What happens at various levels of physics?

  • Subatomic level - Can subatomic particles occupy the same space at the same time. What if they did - is there a nuclear explosion? Do we synthesise different elements?

  • Human-scale level - Will the mountain whose rock is presumably already under great pressure be able to contain the extra matter by compression?

  • Mountain-scale level - Will the arrival of the steel sphere cause the mountain to fracture or even explode?

EDIT

A number of answers have doubted the possibility of the object even reaching the destination. In my mind there has been some development that allows a macroscopic object to quantum-jump to the new location. It's the subject of a different question so I didn't want to mention it here. This note is only FYI and I don't intend it to invalidate any answers so far. For now I'll accept the transmission systems (or impossibility thereof) of the answerers.

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I think we have enough information to make a pretty good scientific estimate of what happens, given the unscientific premise.

The first key point to remember is that matter is almost entirely empty space. The mass of matter is confined to the atomic nuclei which are very, very tiny. In solid matter, atoms are separated by about 10-8 meters. Nuclei 10-15 meters across. This means that the fraction of space occupied by nuclei is about 10-7 (the ratio of the two linear dimensions) cubed or about 10-21. (Think of an atom as a 1 cm sphere of very dense nuclear matter in the center of a 100 km sphere of electron cloud. To the same scale, the rock and the metal sphere are essentially a solar-system sized spheres filled with the gossamer 100 km bubbles just touching each other.)

That means that only a tiny, tiny number of the nuclei in the meter sphere of steel will teleport into the same space as a nucleus in the rock. There will be some, so there's still energy release due to nuclear reactions, but it's small, basically chemical-sized. (Think of a meter sphere of dynamite: You don't want to be standing next to it when it blows, but it's not going to destroy a mountain.)

The volume of each atom that isn't nucleus is occupied by the atom's electron cloud. Electron clouds are compressible and the energy created by teleporting in the steel sphere's electron clouds -- you do realize that your problem requires the energy released to come from somewhere? -- is roughly the same as the energy needed to compress a 1 meter steel sphere and and a 1 meter sphere of rock into a single 1 meter sphere of stuff. (The strength of materials -- steel, rock, human tissue, whatever -- is entirely due to the mutual interactions of the constituent atomic electron clouds.) If you take two spheres and compress them into one sphere, you need pretty much the same result energetically as you get from the teleportation.

It should be possible to estimate this, and I'm going to research it. It's important to note that the energy levels involved are chemical (though extreme!) rather than nuclear.

My guess is that this will produce a bigger bang than the accidental nuclear superpositions do, but still not enough to destroy the mountain.

...However, until I could get a solid numerical estimate, I think I'd make a point of watching it from ten or twenty miles away...

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  • $\begingroup$ I agree with the notion of this answer, but I find it difficult to understand how you're phrazing it. If I didnt already know about the "emptiness", I don't think I'd've understood this. I suggest rephrazing and maybe add an image. $\endgroup$ – Martijn Nov 27 '18 at 15:11
  • $\begingroup$ @Martjin: Good point. A picture doesn't begin to give an accurate idea of how empty an atom is, but perhaps the scale model I described will. $\endgroup$ – Mark Olson Nov 27 '18 at 15:21
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    $\begingroup$ It’s probably worth noting that those nuclei which do appear close enough together will undergo an immediate fusion reaction. It’s not necessary that they appear right on top of each other - the nuclei just need to appear close enough that the strong nuclear force exceeds the Coulomb barrier, much as quantum tunneling might do. Also, fusing elements above iron is actually an energy intensive process, not an energy producing one. $\endgroup$ – Dubukay Nov 27 '18 at 16:08
  • $\begingroup$ @Dubukay: Since one of the two spheres was described as essentially 100% Fe, fusion reactions will always include one Fe nucleus and one other. Thus they will be 100% energy consuming. The energy release doesn't come from fusions, but from two nuclei actually physically overlapping which should yield something like a hyped-up fission reaction -- the nucleus will almost certainly rapidly disassemble. $\endgroup$ – Mark Olson Nov 27 '18 at 16:19
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    $\begingroup$ You can get a lower bound on the energy release by treating the iron/steel mix as an infinitely-compressible spring. Assuming I haven't botched the math too badly, teleporting a cubic meter of iron into a cubic meter of stone will release around 200 GJ of energy, or around that of 50 tons of TNT. Not nuclear-weapons level, but not far from it. $\endgroup$ – Mark Nov 28 '18 at 0:46
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Is there a nuclear explosion? Do we synthesise different elements?

Fission requires free neutrons to interact with heavy elements. This will not happen because the matter you are teleporting will not have a significant number of free neutrons.

Fusion requires two atoms to interact with enough energy to overcome the electro-static repulsion of the atoms nuclei. In this case, your teleporter has no mechanism to add the required MeV of energy to each atom. This amount of energy would turn whatever you were teleporting into a plasma, anyways, so safe to say there is no mechanism for a nuclear reaction take place.

But....

If an atomic nucleus randomly appeared near another atomic nucleus, they would be repelled by the electro-static force between the two positively charged nuclei (the nucleus has only positive protons and neutral neutrons, so all nuclei repel).

Atoms are almost entirely empty space. Let us take the example of one mole of iron at room temperature. This is 56 g and occupies 7100 mm$^3$. The number of atoms in this mol is Avogradro's number. Each atom occupies a volume of 7100 over Avogadro's number. This works out to about 12 cubic Angstroms.

Solving for the nuclear density of iron, we find the radius of an iron atom is about $5\times10^{-5}$ Angstrom and the volume of the nucleus of an iron atom is about $1\times10^{-12}$ cubic Angstroms. If you plop two pieces of iron into the same space with magic/teleportation then there is a 1 in a trillion chance that the two iron nuclei will collide.

Still, for every mole of iron (which is about a trillion trillion atoms), this is still a lot of atomic nuclei that collide, something on the order of a trillion. This collision means that there will be created oddly shaped conjoined nuclei of two iron atoms. Now, iron specifically, is the most stable nucleus in terms of binding energy per nucleon, but other elements are not. Atoms smaller than iron might 'fuse' by being so close that they Strong force is able to overcome electrostatic repulsion, but the odd shape of the almost-close-enough atoms might cause some atoms to be ripped apart.

So in that case, I'd suggest that for solid materials, something on the order of a trillion atoms per mole of material will undergo these unusual reaction. What happens in each reaction depends strongly on what the elements are, the relative stability of their proton-neutron arrangements, etc. So, some funny business will happen, and there may be new elements synthesized and some fusion type reactions that release energy.

How much energy?

A hydrogen fusion event releases something like 30 MeV. Lets say that we release 10 MeV of binding energy per nuclear collision due to whatever various interactions take place. Multiply by a trillion and convert to joules and we get on the order of 1 joule released per mole. That isn't a lot! Unless you have a chain reaction starting (which you probably won't, without free neutrons or a very hot plasma), you actually won't do that much damage to what you are teleporting.

What about a gas?

Since you mentioned gas in the beginning, it is worth noting that since the density of a gas is about 1000 times lower on a molar basis (as compared to a solid), teleporting someone into a gas will cause about 1000 times less nuclear collision, perhaps a billion per mole of teleported object. Still plenty to cause weird physics, but a lot less energy released.

But, looking at our energy numbers, we now have something like mJ of energy released per mole transported. Assuming a person is 5000 mols of water (which is around 90 kg), then you only see about 5 J of energy released by teleporting a person into the air. Since a human generates about 0.5 W of thermal power, this is equivalent to the heat you give off in about 10 seconds.

Now, no promises about what this does as far as DNA damage, but it is almost reasonable to say that you could teleport a human into air without having them experience unusual effects, despite the 5 trillion nuclear collision that you just caused.

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    $\begingroup$ I love science: 5 trillion molecular collisions -> 1 Joul of energy. $\endgroup$ – Draco18s Nov 27 '18 at 18:34
  • $\begingroup$ On your last paragraph: 1 Joule of absorbed radiation = 1 Grey of radiation dosage. If we assume that these funky nuclear reactions are similar to getting hit by nuclear radiation, this is 5 Grey your person has received by teleporting into air. 5 Grey is not benign but actually quite nasty, with a 50% chance of killing you from acute radiation poisoning. Radiation. They won't explode or combust, but they will suffer slower nasty effects. $\endgroup$ – BBeast Nov 28 '18 at 5:10
  • $\begingroup$ @BBeast The problem with that is there is no guarantee the 1 J is all ionizing radiation. It all depends the manner in which the energy gets released; if it ends up being lots of low energy photons, then it just causes heat addition. If the energy all gets released as X-rays or gammas then, yes, the person would have immediate radiation sickness. But I have no idea how much of it would be dangerous radiation. $\endgroup$ – kingledion Nov 28 '18 at 5:26
  • $\begingroup$ @kingledion You assume nuclear reactions. Nuclear reactions emit high energy particles, such as gamma rays. It won't all be gamma rays; you'd have neutrons, neutrinos, alpha and beta particles, fission fragments and more exotic particles too, but you won't get 10 MeV of IR radiation. $\endgroup$ – BBeast Nov 28 '18 at 7:28
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According to some interpretations of quantum field theory, or QFT, the fundamental state of matter isn't particles but fields instead. What we could call a particle is merely a local excitation of a field, or of a few fields that interact with each other.

This requires that the universe be comprised of fields that can have different local levels of energy. It also implies that the universe is permeated in non-zero levels of energy, because any point in the universe that can contain energy isn't empty.

In physics there is a distinction between a vacuum and a void. A vacuum is the absence of particles, or with a QFT interpretation the absence of local excitations of fields, whereas a void is the absence of energy. A vacuum is not a void, as it holds what we call Zero Point Energy.

This Zero Point Energy is just the resting state of a non-excited field. If you were to add energy to this field you would have something that looks like a particle of that field's type. A photon in an electromagnetic field, an electron in an electron field, etc. These fields can interact with each other, an electron can absorb and later emit a photon.

I suggest that instead of viewing it as a collision of atoms, you interpret the teleportation process as being additive. Each of the fields and their energy levels in the mountain ranges suddenly gain the amounts of energy contained within the steel sphere.

This would have a number of incalculable consequences. The simplest solution would be to say that the energy that was added is immediately expelled from the system due to it no longer being stable in the context of another field already containing different energy levels. So all of the energy being added is forced to spread out evenly in the surrounding space. Which is just a very odd way of describing an explosion. Just how big of an explosion would it be though?

The composition of steel is variable but let's say it's 99% iron and therefore has a density of nearly 7 874 kg/m³. A 1 meter diameter sphere has a volume of V = 4 ⁄ 3 × π × R³ = 4 ⁄ 3 × π × 0.5³ = 0.52 m³. So that's a total of 7 874 × 0.52 = 4 095 kg of iron atoms. To calculate the energy contained in 4 095 kg of iron we can apply the famous E = mc² E = 4 095 × (3 × 10^8)² = 3.68^20 J = 0.36 Zettajoule

It's almost the equivalent of the entire world's energy consumption in 2010. For comparison to something we have more direct evidence of the destructive effects of, the atomic bomb that annihilated Hiroshima had an energy yield of a few Terajoules (10^12 J).

In fact it's such a massive amount of energy that it's over 250 times the 2011 earthquake and tsunami in Japan(https://en.wikipedia.org/wiki/Joule#Multiples).

Of course, a non-violent alternative is to imagine that the effects of adding energy to a field changes the composition of the elements we can observe, and transmutes the mountain into various other elements higher up in the periodic table. So instead of rock composed of oxygen and silicon you might have a mix of zirconium and molybdenum. This type of effect, if discovered, could lead to the birth of a transmutation industry. Perhaps it should be destroyed to protect the economy, or used in secret to the profit of a few.

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    $\begingroup$ I love this answer, you deserve a bounty. $\endgroup$ – Renan Nov 27 '18 at 16:08
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    $\begingroup$ I don't think this is good physics. I don't know why you would interpret this as additive energy levels. You don't propose any mechanism for transforming the iron matter into energy, so why would you count the matter energy equivalent values? $\endgroup$ – kingledion Nov 27 '18 at 16:18
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    $\begingroup$ I also don’t understand why you annihilate the iron sphere. E = mc2 would be the case if we teleported antimatter into the mountain, not if we added the fields together. Unless you’re arguing those are equivalent? $\endgroup$ – Dubukay Nov 27 '18 at 16:59
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    $\begingroup$ @Dubukay if you do the sum of all the different types of energies in the different fields I think it's incredibly unlikely to land on anything stable. Meaning if you sum the different quark field energies, electron field energies, gluon field energies (and so on...) of the mountain rock and the iron sphere you would more likely end up with unstable energy configurations that are dispersed rather than something stable. Of course, if you do get something stable I suggest some sort of transmutation effect. Or maybe it would be something in between. $\endgroup$ – Thymine Nov 28 '18 at 8:50
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    $\begingroup$ Physics aside, I like the idea of transmutation when it collides with unexpected matter, this is very much "How scientists accidentely found something that became a game-changer in the world". $\endgroup$ – kikirex Nov 28 '18 at 12:21
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Conservation of energy?

Immediately after this mistake, you have 2x the amount of atoms in a given volume than would normally occupy it. The pressure and temperature will be very high (I think for plasma values of high). Then it will explode. Mark Olson's answer covers this well.

However, where did that energy come from? If you want a science-based answer, you pretty much have to maintain or explain violation of conservation of mass-energy. So:

  1. It physically can't happen. A teleport puts in just enough extra energy to cope with a few stray air molecules in a high-grade vacuum at the specified destination. The object or person being teleported does not explode. A lower-grade vacuum might cause the equivalent of non-lethal radiation sickness or (commonly hydrogen) embrittlement of metals. You simply cannot insert into high-density matter by mistake. The military, of course, can use the energy of an H-bomb at one end to release the energy of an H-bomb at the other ...

  2. The object can be de-materialized but not re-materialized. It gets stuck in hyperspace or wherever, and lost forever. Or it follows the path of least energy to the nearest good vacuum and rematerializes in orbit. (Rather closer to the sun, to deal with gravitational potential energy). Teleportational tunnelling might be the appropriate term here.

  3. Your universe doesn't have conservation of energy. This has consequences, which may actually be useful in an SF setting. The laws of your universe are not time-invariant. (Noether's theory). Or, Lagrangian formulations are invalid. Don't ask me what that means in practice.

  4. Heisenberg compensators or other handwavium, give up on hard science beyond making it possible to suspend disbelief.

Note with all three cases, gravitational potential energy and planetary rotational velocity must be accounted for. Also there are conservation of momentum issues, and hence in case 3, the laws of the universe are not space-invariant either.

Vernor Vinge was playing with 3. in his Zones universe, but I don't think there was anything that would withstand mathematically rigorous scrutiny.

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Comic book teleportation does not exist and is not supported by physics, so the short answer is nothing of that can happen.

Supposing you want to force the issue, though, you need to define how that matter is going to get from point A to point B.

  • Quantum tunneling is out of question - it only works for subatomic particles, and is too random to teleport a whole body.

  • A big enough wormhole would relocate the whole Earth in a very funny way.

  • Emitting each particle so that they travel as beams, stopping at the destination(a la Star Trek), would probably cause them to not bind to each other again as before. Mind you, each particle might also hit whatever is at the destination as a cosmic ray, which may open a crater on the target mountain of the question.

There is no way to solve this with the tag. If you go with , which is the only way to teleport something as seen in videogames and movies, then anything goes; I personally like John Carmack's solution. Faced with the same question about three decades ago, he invented the concept of telefragging.

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    $\begingroup$ Quantum tunneling isn't limited to subatomic particles, although it is more prevalent at that scale due to the energy levels involved. The size of the object and the barrier is unlimited. Everything else is just a matter of engineering. $\endgroup$ – Skek Tek Nov 27 '18 at 17:19
  • $\begingroup$ If you want to teleport safely, the best solution is to swap the contents of two volumes of space. $\endgroup$ – Mark Nov 28 '18 at 0:26
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Fair warning: this answer is a bit of a cop-out.

Still with me? Alright, good. One of the more important things to you can do to create an internally consistent magic system is to follow the laws of physics. Newton's third law states that for every action there is an equal and opposite reaction, and this should be true for magic too. Ok, why am I telling you this? Well, what would be the opposite reaction for teleportation? Presumably, more teleportation. The best way I see to deal with this question is to say that teleportation doesn't just move object A to space B. It also moves whatever was in space B to wherever object A was. That way no matter what happens, there's nothing to collide with when object A arrives.

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    $\begingroup$ The question specifically says "Assume the teleportation is one-way so the rock stays where it is." $\endgroup$ – Thymine Nov 27 '18 at 16:40
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    $\begingroup$ The opposite and equal reaction in magic would be an evil steel sphere. $\endgroup$ – hyperion4 Nov 27 '18 at 17:20
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    $\begingroup$ @hyperion4 You mean a steel sphere with a goatie? $\endgroup$ – Hagen von Eitzen Nov 28 '18 at 8:33
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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.

Though experiment

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.

Force

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.

Newton’s laws

First Law

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).

Second Law

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.

Third Law

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?

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.

Released energy

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.

Other considerations

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.

Addendum: Quantum-jump

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.

Tl;dr

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)

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  • $\begingroup$ Hey, you didn't detail the Einstein equation but I think you may have accidentally forgotten to convert the kg to g (I did the same calculation earlier and have something 10^3 larger than your answer) $\endgroup$ – Thymine Nov 28 '18 at 12:47
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    $\begingroup$ @Thymine double checked, I cannot find any error. I saw your computations, why are you converting to grams? The Joule is defined as kg*m^2/s^2, you get m^2/s^2 from c^2 that you put as (3 × 10^8)²; then you want the mass in Kilogram to get Jouls. I think yours is the one wrong. $\endgroup$ – Theraot Nov 28 '18 at 13:07
  • $\begingroup$ Ah yeah you're right, I foolishly converted to kg to g from assuming everything needed to be the "fundamental" unit in that equation. Thanks! $\endgroup$ – Thymine Nov 28 '18 at 13:14
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If you want to observe the laws of science, then you must explain how a stream of data can be reconstituted at a remote location without a high energy assimilation machine turning that data into a cohesive mass. Energy will not simply turn into mass, much less cohesive arrangements of steel alloy, without that assimilation machine. But if you have such a machine, then it will be responsible for creating a vacuum in which the assimilated mass does not collide with any other objects, not even air, so the coordinates of the reconstituted steel sphere are irrelevant.

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