A more interesting application of this would be to simply "travel" to a distant future on Earth ("traveling to the future" is, I think, a bit of a misnomer, because we are technically speaking constantly doing so; doing so in a different frame of reference is no more of "traveling to the future" as anything else.)
In other words, accelerate at 1g away from Earth for a given time (like for a year) then decelerate and come back. Total travel time for the travelers is four years, but much longer will have passed on Earth. The travelers will be arriving to a relatively distant future Earth, being themselves only four years older.
I'm pretty certain that many, many people would be completely willing to do this. (Although it would of course be a risk. They could just as well arrive at an Earth that has been completely devastated by a new world war or some other catastrophe. Something that would not have happened in their own lifetimes if they have stayed.)
Hello everyone. I suggest you quite interesting challenge.
WARNING: I beg you, please don't google for answer. It's known everywhere. Also, if you know already, during reading question, don't participate. Ok? Challenge is exactly in checking your own skills.
Now. The question.
Describe, why it behaves like that:
https://scontent-ams3-1.cdninstagram.com/t50.2886-16/15719832_950998775031904_6513587924682932224_n.mp4
I just stumbled across it today. And, was wondering that it's actually kinda hard to prove this behavior should occur.
Suppose the distance between two objects in space is 100 light-seconds, but space is expanded so that every second another 100 light-seconds is added to this distance. Would a photon sent from one object towards the other reach that other object?
Let's simplify the problem by making the expansion happen in discrete steps. In other words, space is static, and we wait for a second for the photon to travel, then we expand space in one discrete step so that the distance between the objects increases by 100 light-seconds, wait for another second for the photon to travel, and so on.
During the first second the photon travels 1% of the distance. During the next second it travels 1/2% of the distance. Then 1/3%, then 1/4% and so on.
As we know, the sum 1+1/2+1/3+1/4+... diverges, which means that the photon will eventually reach the other object. But how long does it take? (I know that "how long" is a complicated question in general relativity, but I suppose it's "from an external point of view".)
Of course in actuality the distance between objects (eg. galaxies) doesn't increase linearly. A constant rate of expansion of space causes the distance to grow in an accelerating manner (the distance between two objects doubles every n seconds, which means that the growth of distance is exponential).
Is there a limit for a constant rate of metric expansion of space that still allows light to reach receding objects that start at a given distance from each other? Is there a formula for this?
Suddenly, sometimes discrete is harder than continious.
Here is solution for continious task:
Assume photon starts from coord 0 and travels to coord 100.
Then, Norm is changing ||x|| = |x|*(1+t). i.e.
t = 0: 1+t = 1
t = 1: 1+t = 2
t = 2: 1+t = 3
So expansion same as in task.
Then, speed in terms of coords is 1/(1+t), and time it will take is
integral of 1/(1+t) dt from 0 to x.
So, equation is:
log(x+1)-log(1) = 100
log(x+1) = 100
e^log(x+1) = e^100
x+1 = e^100
x = e^100 - 1
Here, and everywhere above, log is natural logarithm.
It's curious how that's something like 77 orders of magnitude larger than the age of the universe, yet from the photon's own perspective the travel time is instantaneous (at least assuming it can travel in vacuum the whole distance).
This seems paradoxical. The universe could end before the photon reaches its target, yet from its own perspective it reaches the target in 0 seconds.
This seems paradoxical. The universe could end before the photon reaches its target, yet from its own perspective it reaches the target in 0 seconds.
For the photon (or all massless particles), from the start of the universe to the end of the universe, everything happened in 0 seconds. In a way, mass allows us to experience time.
Saying the universe would end when there is still a photon moving around doing stuff depends on your theory on what counts as the world having ended.
Warning: Might glitch to creditsI will finish this ACE soon as possible
(or will I?)
Of course a much more mundane and closer-to-life example would be: A photon is emitted from the Sun. At that exact moment it has an unobstructed path towards the Earth. However, while it's on its way, something gets in its way, blocking it. Thus it hits the blocking object rather than arriving at the Earth.
But from the photon's own perspective this all happened in 0 seconds, which seems paradoxical. The moment that it left the Sun, and the moment it hits the obstruction, are the exact same moment. (The distance between the Sun and the object is zero, no matter how far apart they are from an sub-lightspeed point of view...)
It could also happen the other way around: When the photon is emitted, there's an obstruction between it and the Earth, but right before it hits the obstruction, it moves and the photon doesn't hit it and arrives at the Earth instead. But from the photon's own perspective there wasn't any obstruction to begin with. Which seems paradoxical.
On another topic, it's estimated that the photons that arrive to Earth from the Sun are actually tens of thousands of years old, because it takes that much for those photons to reach the surface of the Sun from its core.
But this seems to imply that every single photon created in the Sun is created in its core. One would think that eg. the solar corona emits boatloads of photons itself. Is it really true that all photons from the sun are tens of thousands of years old, or would it be more accurate to say that some of them are?
Of course a much more mundane and closer-to-life example would be: A photon is emitted from the Sun. At that exact moment it has an unobstructed path towards the Earth. However, while it's on its way, something gets in its way, blocking it. Thus it hits the blocking object rather than arriving at the Earth.
But from the photon's own perspective this all happened in 0 seconds, which seems paradoxical. The moment that it left the Sun, and the moment it hits the obstruction, are the exact same moment. (The distance between the Sun and the object is zero, no matter how far apart they are from an sub-lightspeed point of view...)
The emission of the photon and the the imposition of the obstacle are separated by, to use a bit of jargon, a spacelike interval. (Spacelike means the distance between the two events is greater than the distance light could travel in the time between them). Spacelike intervals have two key properties: they are not related by causality, and the chronological order in which the events happens is frame-dependent. So, you are right that the light ray feels no time at all in its own frame, but, from the light ray's perspective, the barrier is put in before it sets off - which resolves the paradox.
On another topic, it's estimated that the photons that arrive to Earth from the Sun are actually tens of thousands of years old, because it takes that much for those photons to reach the surface of the Sun from its core.
But this seems to imply that every single photon created in the Sun is created in its core. One would think that eg. the solar corona emits boatloads of photons itself. Is it really true that all photons from the sun are tens of thousands of years old, or would it be more accurate to say that some of them are?
Most of the photons are emitted deep within the core, simply because it's hotter there (over a thousand times hotter in the sun's core as compared to its still-very-hot-by-everyday-standards surface), and you need those incredibly high energies to power the nuclear fusion within the sun (which is what emits the photons).
There might be some emitted from nearer the surface, from things like the decay of free neutrons, but I would imagine the proportion would be insignificant.
This seems paradoxical. The universe could end before the photon reaches its target, yet from its own perspective it reaches the target in 0 seconds.
You just underestimating rate of expansion in your statement.
I don't know how to explain it.
But expansion rate is close to adding same length as being passed in single turn.
Intuition tells to some people that light should not reach end at all.
Your intuition is completely opposite: you're thinking about speed of light, and how short distance is.
So, none of intuition above works well. But first one is close to real situation.
Trick here that any move towards end, is changing ratio of passed distance to total distance. In other words, it increase progress in percents.
It isn't proof either, because it may grow infinitely long, and never reach 100%
as in case when each second space extends in twice.
Future time travel is easy to understand i.e. exceed the speed of light and time will speed up relative to you in order to compensate. However, I've never understood how time traveling into the past in theory works (never mind all the possible paradoxes that could bring). Despite physicists saying it's theoretically possible to do.
Future time travel is easy to understand i.e. exceed the speed of light and time will speed up relative to you in order to compensate. However, I've never understood how time traveling into the past in theory works (never mind all the possible paradoxes that could bring). Despite physicists saying it's theoretically possible to do.
Try to get your head around the fact that, theoretically, according to general relativity, a particle could travel to the past (using the ergosphere of a black hole) and collide with itself.
However, I've never understood how time traveling into the past in theory works (never mind all the possible paradoxes that could bring). Despite physicists saying it's theoretically possible to do.
There are solutions to the formulas of general relativity which contain closed time-like curves, and those would allow at least some form of backwards time travel.
One solution with closed time-like curves involves wormholes. Wormholes cannot just connect different points in space, but also different points in time (if they don't, just accelerate one end, and time dilation does the rest). Now if the time difference is huge, and the space distance is small, then it's possible to enter the "future" end, arrive at the "past" end, then travel outside the wormhole back to the "future" end. If you're fast enough, you can arrive before you left, meeting younger yourself, and then you get to have sex with your grandma or something (I forgot how these work).
All these theorized forms of time travel involve regular travel along weird spacetimes. There's no machine with a dial you can set to "hitler's birth", and *poof* you're there - never mind the fact that such a machine would get the location wrong, due to the earth constantly moving. If you have a sufficiently weird spacetime, then your time machine is a regular space ship, and the journey will take time (i.e. the travelers will age).
However, it is not at all obvious if such solutions describe (parts of) the universe we live in.
For one, the fact that something would not violate GR does not imply that it exists. More intuitively, a skyscraper made of unicorn-flavored pudding does not violate GR. That doesn't mean that one exists, it doesn't even mean that one can be constructed, and it certainly does not imply that such a thing would be stable. Same with wormholes.
Second, GR is not an accurate description of our universe, because it ignores quantum effects. As far as I'm aware, all solutions with closed time-like curves contain parts where quantum effects are expected to be significant. We'll have to wait for a theory of quantum gravity, and then re-run all those solutions. Many physicists expect the closed time-like curves to go away when including quantum effects.
You can also start at the other end, QM. The schrödinger equation is pretty clear on the concept of time, and that does not involve time travel. But it's not a complete and accurate description of our universe, either, so we still don't know.
tl;dr: backwards time travel is really unlikely, but has not conclusively been proven impossible. GR says "maybe", QM says "no", and at least one of those answers is wrong.
For those who want to remain hopeful about time travel, it's worth remembering that when Karl Schwarzschild first did the maths that predicted the existence of black holes, he expected them to be solely a theoretical curiosity that would not actually exist in our universe. He was wrong about that, so maybe we're wrong about wormholes, too?
I guess the difference is that it's a lot harder to see a mechanism for how a wormhole might develop. A black hole is just large amount of matter which has collapsed under its own gravity - simple. But a wormhole is a far more complicated structure.
You don't need a wormhole to allow time travel. It is my understanding that according to the equations two rotating black holes close enough to each other (which pretty much certainly happens when two black holes, which existence is in very little doubt anymore, are in the process of colliding) will cause a situation where a particle could travel through their ergospheres in such a manner that it will arrive back at its starting point before it started.
I have no idea what the consensus is if a particle does indeed do that, in real life. For example, what happens if the particle does that, and collides with itself? Perhaps there even isn't a hypothesis about this.
A table tennis ball has a diameter of 40 millimeters, and a mass of 2.7 grams. Two of them are floating in empty space, at rest relative to each other, 10 meters apart, with no other significant forces affecting them.
How long does it take for them to collide, and what's their relative speed when they do so?
A table tennis ball has a diameter of 40 millimeters, and a mass of 2.7 grams. Two of them are floating in empty space, at rest relative to each other, 10 meters apart, with no other significant forces affecting them.
How long does it take for them to collide, and what's their relative speed when they do so?
This is a classic problem and I'm rather proud of my treatment of it. When I was in graduate school, everyone else in the class just pulled the answer from an online document which had a step that stated, "The above integral can be evaluated as..." Well that's no fun. My solution sidesteps the calculus, which appears in an earlier derivation.
It can be shown using Kepler's laws of motion that the time it takes for two bodies that have a 1/r^2 force dependence to complete one orbit is
where a is the semimajor axis, k is the force coefficient, and mu is the reduced mass:
For gravitational attraction, we have
and so our original equation reduces to
So how is this useful when the objects aren't orbiting and instead are gravitationally attracted directly toward each other? Well, a collision-course is just an elliptical orbit with an eccentricity of 1. Furthermore, the period of the orbit can be split into four different sections that take equal time, assuming the objects can pass through one another: the objects are released, then they "collide" (pass over the same point), then they swap places, then they "collide" again, then they return to their starting point. We're therefore solving for
For the values you supplied (a = 5 m, m_1 = m_2 = 2.7 gm), I calculate 2.93*10^7 seconds or 339 days. Not too shabby-- I was expecting a longer length of time. This is a close approximation to the real answer, which will be slightly less due to the finite radius of the balls. Before we improve the answer, we'll need to calculate the final speed of the balls.
If the balls had zero radius, their speed as they pass through each other would be infinite. Instead, we'll need to conserve energy to find their final velocity:
where m is the mass of either ball, R is their initial separation, d is their diameter, and v is their final velocity. On the left hand side is the change in the gravitational potential energy and on the right hand side is the sum of the kinetic energies of the two balls.
Rearranging this expression, we find
which evaluates to 2.11*10^-6 m/s. Each ball has 20 mm left to traverse until they are directly on top of one another, which at their final rate of travel wold take 0.109 days (2 hours and 37 minutes). (It's actually a bit less than that because the balls would continue to accelerate if they didn't collide.) This is insignificant compared to the 339 days that they had already been traveling.
Finally, I'll suggest that this answer is based on the incorrect assumption that there will be no electromagnetic forces between the two balls. It would not surprise me if celluloid (which is what table tennis balls are made of) is slightly diamagnetic. This would give rise to a slight magnetic repulsive force between the balls that may not be negligible compared with the gravitational force. These kinds of problems-- with two spherical objects exerting a magnetic force on each other that further polarizes them-- are actually quite difficult and an approximation of their solution was only found around the mid-'90s. As a gut feeling, I expect the magnetic force contribution to be negligible, but I'd like to put pen to paper to confirm it.
As we know, the sum 1+1/2+1/3+1/4+... diverges, which means that the photon will eventually reach the other object. But how long does it take? (I know that "how long" is a complicated question in general relativity, but I suppose it's "from an external point of view".)
Of course in actuality the distance between objects (eg. galaxies) doesn't increase linearly. A constant rate of expansion of space causes the distance to grow in an accelerating manner (the distance between two objects doubles every n seconds, which means that the growth of distance is exponential).
Is there a limit for a constant rate of metric expansion of space that still allows light to reach receding objects that start at a given distance from each other? Is there a formula for this?
Your observations are not paradoxical at all. In relativity, time passes differently for different observers, that occurs even in flat space. In order to ask how long something takes, it is necessary to specify the frame where the measurement is made.
And yes, there is a limit to how fast the universe can expand so that all points are reachable. Current observations indicate expansion happened pretty fast after the Big Bang. It is an open problem in cosmology to explain how the universe is so homogeneous if its parts could not communicate on primordial eras.
