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I wonder if a time machine can be the cause of a bootstrap paradox?

Background: Let's say one day I found in my room a portable time machine. I use this machine to travel to the future, where I realize I am the person who placed the machine in my room. I then travel to the past to drop off the time machine and let my past self use it.

The time machine always travels with me. There is only one time machine.

Condition: Now let's say this time machine has critical components built of iron and, after many, many iterations, it rusts and falls out of order. At some point I won't be able to travel to the future ... but the time machine should be here, shouldn't it ?

Question: Is something wrong with my scenario, or could this happen? (If so, how is it possible (theoretically)?)

The paradox only occurs if you assume a single timeline. In which case it is pretty much identical to the 'grandfather paradox'. In the left fig. below, if there is a single timeline then there is a single 'time-path' for you (red line) - that doesn't in itself produce a paradox, but the time-machine also only has a single path (blue line) which by definition must be closed. So the time machine would age further in each loop and result in your paradox - unless time travel itself somehow reverses entropy in the machine in which case you could arm-wave yourself out of the paradox.

If you accept a multiverse-like multiple history situation, things are a little more complicated but there is no paradox. On the right below, some initial you (red line) gets access to a time machine, travels forwards and then backwards. On arrival at the end of the backwards step the timeline splits. The original timeline continues (in which you disappeared, reappeared in the future and then disappeared again forever). An alternate you (orange line) now repeats your journey in a first alternative universe,. This can be repeated numerous times until eventually (with the yellow-line you) the time-machine breaks and yellow you, the time machine, and your green-you alternative spend their lives all together in a final alternative. Note that the blue line is open , not closed, so the machine can obey the normal laws of entropy.

There is still a potential paradox in answering where the time machine came from in the first place. But a non-paradoxical solution would be that in some 'first' time-line, someone invents the time machine, then someone travels with it back in time and in a 'first' alternative timeline, stops that person from inventing it, so no 'other' time machine appears in any of the observed timelines.

The Bootstrap Paradox is a theoretical paradox of time travel that occurs when an object or piece of information sent back in time becomes trapped within an infinite cause-effect loop in which the item no longer has a discernible point of origin, and is said to be “uncaused” or “self-created”. It is also known as an Ontological Paradox, in reference to ontology, a branch of metaphysics dealing with the study of being and existence. (Source)

Can the time machine be a necessary component of the bootstrap paradox? sure! A really good example is, IMO, the machine in the movie Primer (2004). Does your scenario describe a viable bootstrap paradox? Maybe.

All bootstrap paradoxes focusing on an object have one basic problem: the object never seems to get old. This is because the explanation of the paradox is, IMO, inadequate. Note that the quote above states, "...trapped within an infinite cause-effect loop..." This is NOT a requirement for a bootstrap paradox. In fact, it's almost never the case than an infinite loop exists (at least I've never heard of one where it had to exist, or could even be assumed to exist).

Why? Let's look at your scenario. you-before-first-use finds the machine and uses it, causing you to become future-you. Future-you travels back in time to place the machine. And there's the problem. Future-you watches you-before-first-use use the machine ... and then lives out the rest of his/her life in comfort and profit (having sold the rights to your autobiography). The loop only occurs once. The only way it could occur an infinite number of times is for future-you to always be the one who uses the machine the first time.

Curiously, neither example used in the source I quote above represents an infinite loop, even though the article believes they do. In Heinlein’s short story “All You Zombies” (1959), the protagonist travels back in time to impregnate his/her former self, becoming his/her own parent. OK, the person who traveled back in time didn't disappear. The linearity of time for the traveler is unique with a single loop. It can only be thought of as happening over-and-over if you ignore the future of that aspect of the protagonist (who most likely lived a long and happy life...)

The example of Somewhere in Time (1980) involves a pocket watch. But, once again, the loop only can occur once unless you assume that when Christopher Reeves traveled back in time he somehow merged with his future self.

It's easier to answer the question by asserting a single loop — but then you wouldn't have a question.

Since the loop can only occur once in your scenario, there's no issue with the iron components becoming rusted and failing.

What would it take to get infinite loops in my scenario?

Star Trek the Next Generation investigated two possible ways.

One is like in Star Trek the Next Generation's Cause and Effect. The ship itself is thrown back in time to a starting point. When tracking the linear history of the ship, there is only one ship repeating the same loop in time over and over and over. The show creatively lets the crew in on the secret by providing means of "communicating" outside the loop. There are never, for example, two Picards. There's only ever one. In this case, nothing can age, and so your iron components would never fail. Said another way, time simply "resets" to the beginning of the loop.

The second is STNG's episode Time Squared. In this, a future Picard is thrown back into the past. But this episode cheats the question by never portraying an infinite loop. This is a good example because it shows what to do with the duplicates (Picard and the shuttle craft): they're destroyed in the explosion. So, although the episode does not show an infinite loop, one could be created if Picard took the same actions over and over.

But in your case, you-before-first-use always uses the machine. There's no way to create an infinite loop because you need future-you to always make the choice.

Conclusion

Which returns us to the above stated conclusion: your machine will never take more than one iteration and therefore the component will never wear out. If you figure out how to make iterations, it will either always result in a new machine or in the machine never aging.

There is a thought experiment out there that says that time travel is impossible; not because of the physics, but because of what we do with it.

Basically it goes like this. If time travel is possible, we'll use it to change the past. Eventually, we'll change the past to one in which it is impossible to create or use time travel. Hence, time travel is impossible.

Your antiquating time machine paradox is really just a less complex version of this. The loop is infinite. The durability of your time machine is not. The Laws of Entropy state that this loop has to eventually fail because eventually the machine fails. While the Ship of Theseus model can help, the problem is that every time you consume a resource in the past to fix the machine, you're also making a minor change to the past that could (eventually) accumulate to a future which precludes the use of the time machine. Even if you only use parts in the future, or the latest point in your loop, eventually the supply of those parts has to run out also. All things considered, this is not an infinitely tenable scenario.

If you're looking for a way to do this, then Ship of Theseus is the way to go. That is, the Time Machine is made up of parts and the parts are gradually replaced alongside this time-travel cycle. Each part is replaceable, but despite that, the ship cannot be completely built from scratch for reason which aren't understood. Of course, this doesn't really help because any time loop (which puts physical objects in the past) generates entropy and these time loops are supposedly infinite, which means an infinite increase in entropy.

You're asking 'Is this possible if I don't replace it?' and the answer is 'No', but it's not 'No' because the machine rusts, it's 'No' because it's a self-fulfilling paradox. Time is linear, not circular, the time machine can't be there in the first place. Something cannot come from nothing.