Time curvature and twin paradox.

Imax

Can the twin paradox provide us with insight into time curvature?

If my twin boards a ship that can travel near the speed of light, special relativity says that on arrival back on Earth, my twin should be younger that I am. Has my twin experienced a time curvature?

tiny-tim

Hi Imax!

No such thing as time curvature.

There's space curvature, and space-time curvature, but not time curvature on its own.

Also, the twin paradox can happen in special relativity, in which even space-time is completely flat!

Chalnoth

Although, if you want to be pedantic, you have to accelerate in order to get the twin paradox, and special relativity can't handle accelerations.

tiny-tim

it jolly well can!

it can't handle accelerating observers, but it can certainly handle accelerating clocks, which is all the "paradox" needs

Chalnoth

Hmmm, my understanding was that was just an approximation, though. Still, I suppose it is an incredibly good approximation for any amount of acceleration that humans are likely to survive.

Radrook

The faster you travel the greater the discrepancy between the time as it relates by us and another creature undergoing a different velocity. An accelerated clock slows down and subatomic particles last far beyond their usual lifespans when undergoing close to light velocities . Both phenomenon have been experimentally proven.

Based on Einstein's theories and these observations, it is thought that if a human is subjected to speeds approaching light then his perception of time passage will differ from those on earth because his body will be aging at a slower rate and all body functions will be slower from an observer's perspective. Going to a planet which is light years away then, might seem to him as a matter of months. But to those on earth it could be a decade. That's one reason that makes traveling at such speeds impractical.

Also, it is assumed that living things would survive that phenomenon. Such an extrapolation from the inanimate to the animate, however, decreases it's certainty factor considerably.

Imax

Space-time can have a curvature, which implies that time may also have a curvature. I'm still struggling with this idea, but if you start out with two synchronized clocks and after doing something, those clocks are no longer synchronized, then it could be that time has a curvature.

Another example. If we can send you in spaceship into a black hole, then we should observe you slowing down as you approach the event horizon. To you, time is not slowing down. In 10 seconds or so you’ll be gobbled up by the black hole, and some of your molecule can form a nice jet about the black holes axis of rotation.

There’s a disconnect in time. I see you slowing down as you go into the black hole, and you could see an acceleration of how the Universe evolves.

tiny-tim

But what can that possibly mean?

Imax

I can’t see time dilation predicted by special relativity in general relativity. How does GR explain time dilation?

Chalnoth

General Relativity reduces to Special Relativity in a flat space-time. So basically, the argument for time dilation is exactly the same, except that now you can also get time dilation, in certain cases, through curvature as well.

BillSaltLake

GR and time dilation example: Photons go radially outward from a large (heavy) planet. They lose energy as they rise. (This energy loss can be approximated even using classical mechanics: use E/c² as the "mass" of the rising photon.) By the time the photons are much higher, their energy is lower. Because E = hf, the frequency f is also lower.
Now suppose the photon source is a radio station broadcasting at a (very low) f = 1 Hz. When the photons reach a distant observer, they may be at 0.9 Hz for example. If every time the crest of the (electric field compenent of) the radio wave is generated, a giant clock facing upward ticks off 1 second, the distant observer will see the radio station and clock taking 1.11 sec per cycle (tick). Ergo, the observer sees time running slow on the surface of the planet.
The twin paradox can be reformulated as the twin circling you at near c. (Assume both of you are in free space and you are inertial, but you twin of course is not.) If a photon had to reach you by "climbing" against centripetal acceleration through a light pipe connecting you to him, the photon would lose energy.

Imax

We seem to live in a compact Universe, with the potential for curvatures in both space and time. By time curvature, I mean to say that two clocks are no longer synchronized. I can’t work out the math :)

Chalnoth

Why would you think it's compact?

Then you're using the wrong word. But no matter which way you slice it, it's not possible to have curvature in only one single dimension. You need at least two to have any curvature.

Fredrik

About SR and acceleration...

If you choose to define the spacetime of SR as the set ##\mathbb R^4## equipped with a set of coordinate systems such that a change of coordinates is given by a member of the Poincaré group, then by choice, the theory only includes inertial coordinate systems. But it can still handle accelerating objects, in particular, as Tiny-Tim said, accelerating clocks.

And there's no reason to define the theory this way. I think it makes much more sense to define the spacetime of SR to be the smooth manifold with underlying topological space ##\mathbb R^4## and the standard smooth structure on ##\mathbb R^4##. This way, a lot more coordinate systems are included, not just "non-accelerating" and "accelerating", but also coordinate systems that don't correspond to measurements performed by an observer.

If we do it this way (and I really think we should), the difference between SR and GR is just that in SR, the metric is always the Minkowski metric (regardless of the matter content of spacetime), and in GR, it's a solution of Einstein's equation.

smyth

Almost thought i understand Twin paradox... but Wiki blew that thought away. Quote from wikipedia: "If the spaceship accelerates at a constant 1g, he will after a little less than a year (mathematically) reach almost the speed of light, but time dilation will increase his life span to thousands of years, seen from the reference system of the Solar System, but his subjective lifespan will not thereby change. If he returns to Earth he will land thousands of years into its future"
Now, is there any difference between acceleration due to gravity, and speed acceleration?I mean aren't we experiencing 1 g in our every day life? from what i understand, the age difference between the twins is due to acceleration, but since both experience 1g how come there still is age delay?

clamtrox

The difference is that one twin is moving while the other is not, just like in the regular twin paradox.

Janus

Acceleration, while it plays a part in determining the final respective ages of the Twins in the Twin paradox is not, in of itself, the cause of the age difference. In other words, it is not the g force that the Twins experience that results in their age difference.