, Just... excellent, as always. You took some time to do the work and I only can appreciate that (I was going to answer in a very concise and probably not very illustrating manner but waiting for this was a better option). Slowly this thread is becoming a reference for some google searches thanks to you and that is awesome, first becuase you are educating people and second because you are making SE known into curious beings. So thanks again for the effort.pzampella
, The question is also very interesting here. Portals might be impossible physics but I get the idea behind the example; would an object falling in a gravitational field archive faster-than-light speeds? If not, where is the gravitational potential energy going? Is kinetic energy even defined in the same way as in Newtonian terms? The problem stated does not need for portals and strange physics to be invoked (and that is what makes this even more interesting). Think about an infinite universe where only Earth and a rock are present. If you let the rock fall down to Earth from infinity it would accelerate until it crashes at a speed of around 11 km/s (the escape velocity of Earth). So you never get to light speed. But! On the event horizon of a black hole you "have" an escape velocity equal to the speed of light. So if your universe had only a rock and a black hole and you let the rock fall from infinity (starting from rest relative the black hole) then all the previous explanation from Watsisname would be similar. Nature does this without portals. The only thing that makes your version of the problem more easy to address is the fact that the acceleration of our rock changes with time while with your portals we can compute for a more or less constant acceleration. But yeah for the rock falling to a black hole all the above graphs would be the same except for a huge stretch in the x-axis at the first part of the plots (not so fancy and explanatory indeed).
, You even managed to create excitement with the next topic (simultaneity in relativity) in a very smooth way. I'm still learning the basics here (I will for many years after my Master thesis) but my prediction is that from the perspective of the observer at rest relative to the portals the act of "entering one of the portals and exiting the other instantaneously" just breaks. Also here, portals show a very strange situation that might show how absurd the idea of portals could really be.
Let's define a portal here as a device that lets you transport instantaneously from one point in space to another. But what does instantaneous even mean? From which perspective?
Let's have 3 frames here. M
is the observer moving through the portals, A
is a frame at rest with respect to the exit portal (the one farthest from the planet) and B
is a frame at rest with respect to the entry portal (the one closer to the planet). Since we are assuming both portals to be at rest relative to one another we know that A
is at rest from the perspective of B
is at rest from A
's perspective also. Great!
So, in our definition of portal. What is instantaneous?
From the perspective of M
instantaneous transport means that there is a time t were the the object is located in two parts of the universe or even better there is a time t-dt
were the oberver is located in one part of the universe and a time t+dt
were it's located in another part.
But this is not instantaneous for the observer in A
in the general case since both portals could have very different clocks (out of sync and even with different flows of time). In fact in this scenario you presented we have a portal closer to the planet and therefore embedded in a slightly stronger gravitational field. As we know from relativity this means that the clock in B ticks slower than the clock in A putting both portals out of sync
as time goes on from the perspective of A
(and something complementary from B
's perspective). For A
, the definition of portal as an instantaneous transport as seen from the perspective of M
would not be instantaneous at all. Since the clock in B
goes slower than in A
this means that A
would reach his time ta
first and B
would reach it later, so when A
sees the object entering the B
portal some time has passed since it went out of the A
portal. The object would be duplicated in the field of view of A
and it would look like if the object had transported to A
before entering to B
(a jump in time). So if we are defining portals with instantaneous transport from the point of view of M
then we are in conflict with portals defined as instantaneous transport from the point of view of A
. If we want instantaneous transport from the point of view of A
we are asking for M
to travel back in time so that the transport is not instantaneous for him. Might be wrong but let's wait for Watsisname to confirm this
With portals we are breaking the continuous aspect of spacetime. I think this violates some of the ideas in special relativity since you would be able to access events outside your light-cone
and therefore you could travel back in time (reversing causality) and also could be viewed as if you were traveling at infinite speed
from one portal to the other. But let's play a little with the idea even if it is probably flawed.
Even if we imagine the portals to be subject to the same gravitational field there still are some disturbing and subtle consequences. To explain this subtle effect erase the planet in this new scenario and think that the 1 m/s2
acceleration is now provided by the rockets of the spacecraft and not by the gravitational field of the planet. This would not make a difference for M since the equivalence principle (also from relativity) says
that all the effects should be the same for a free-fall in the presence of a gravitational field or just by constant acceleration without a gravitational field (I hope to be correct with this, sometimes it's tricky), but it would make a difference for the portals since now they are not subjected to different gravitational forces or any at all and therefore time dilation due to gravity would not be experienced.
Now, from A
's perspective, both A
have clocks that tick at the same time and in perfect sync, so portals could be defined as devices that are able to transport an object from one place to another instantaneously with respect to both A
. Would this definition of portal be valid for the understanding of what is instantaneous for M
? Would M
think of A
as true portals?
This effect we are going to talk here (the break in simultaneity
) does not need for an accelerating object so let's keep it simple. Suppose that M
is moving relative to A
(no difference in this choice) with constant speed. This is the inverse problem of Einstein's train
. Any event happening at portal A
at the same time as any event happening at portal B
from the perspective of either A
(like the object transiting through the portals) would not look simultaneous from the moving object in M
. In fact from M
's view point it would seem like he entered the portal and traveled back in time since time in A
would lag behind time in B
from this perspective. How much? It depends on the velocity of M
(as every effect in special relativity we usually mention) but more importantly it depends on the separation between the portals. If the portals were at sufficiently greater separations M
would see them as if they were at different eons of history and at the time of traversal of B
we would travel back in time. The object would exit A
even before it entered B
. To make things weirder the separation between A
is also relative to M
's motion (as explained by Lorentz contraction
Since here the separation is small (I assume) the effect is very tiny. But, if I'm doing this correctly, the fact that the object crosses the portals many many times could accumulate this effect until it's noticeable (more so if we are close to the speed of light). This effect also happens in the scenario where we were dealing with the gravitational field of the planet but on top of it there was this gravitational time dilation thing affecting the portal clocks. So we have to take into account both things. Also we have to take into account acceleration (my last scenario was with constant speed) so we have to think about this "getting duplicated in time" effect as an increasingly strong effect.
I really have no idea if this works this way since I'm to pedantic to explain my bet but to lazy to do the math and some beautiful graphs as Watsisname. But I now would really like to know if I'm correct with my statements so I will eagerly wait for his response