Into the past ?: Tricky SpaceTime Diagrams?

[Gerinski]

Into "the past"?: Tricky SpaceTime Diagrams?

It's commonly stated that according to GR, traveling faster than light would mean going backwards in time (to "the past").

When I try to picture the situation with the aid of a spacetime (light-cone) diagram, I still don't get why should it lead to "the past".

If we start with 2 events separated by space but on the same horizontal line (let's say, the sun on the left side, and 4 light-years away to the right, the star alpha-centauri). Let's say the horizontal is sun's time 2004.
As the situation evolves in time, both objects go upwards in the diagram parallel to each other. The light emmited by the sun is pictured as a diagonal line at 45º (light-cone), going upwards-right towards alpha-centauri.
The light-cone will reach the vertical line of alpha-centauri at a height of 4 years (sun's time) from the starting horizontal (sun's time 2008).

Now, if the sunlight would have traveled faster than light (the diagonal at more than 45º from the vertical), it would have arrived at alpha-centauri let's say in sun's time 2007, or 2005.
Isn't that still "the future" if the departure was in 2004? Sure it's "less far into the future" than at light speed, but still the future.

Shouldn't it get downwards in the diagram (below the starting horizontal) in order to be considered as going into "the past"? (before sun's time 2004)

I know I'm wrong but I'd like to understand where's the flaw. I guess there is something tricky about trying to visualize this with a spacetime diagram.
Or is it just that 2006 has to be considerd anyway as "the past" even the departure was in 2004?

 

[HallsofIvy]

"It's commonly stated that according to GR, traveling faster than light would mean going backwards in time (to "the past")."

I don't know of any text that states that. What SR texts do note is that Lorentz factor becomes imaginary if v> c and that simply indicates that you can't move faster than light.

 

[selfAdjoint]

Trying to explain this saying to myself (especially the limerick about Miss Bright), I devised a tricky spacetime diagram and proved to my own satisfaction that if you send an FTL^* signal to a ship spacelike related to you and traveling away at a large fraction of c, and the ship echoes the message back at the same FTL speed, it will arrive in your past. I believe I posted this model a long time ago. Maybe Sol has the link.

FTL messaging in a sublight traveling universe is a common science fiction idea. The FTL signaller, under the name "ansible", was dreamed up I think by James Blish and has been used by many, notably Ursula K. le Guinn. But they never deal with the time travel consequences.

^*Usually I used instantaneous communication to simplify the diagram. But I did confirm that any FTL speed works.

 

[robphy]

I am also not aware of any texts that say that. However, this is what the writer of that statement might mean:

Consider one's light cone at event P. If one's worldline is able to travel outside that cone, then there is an event Q on that worldline which is spacelike-related to P. Although one may have (according to his clock) t_Q>t_P, some other observer might have t'_Q<t'_P. (Causal order is not an invariant for spacelike-related events.) [In some reference other frame, t"_Q=t"_P.]

 

[Gerinski]

Thanks !
I think I was also probably confusing statements like "time would run backwards" with "travelling into the past"

 

[Gerinski]

Thanks again, yet just let me reshape my question to clarify what I meant.

I think the statement "for anything traveling FTL, time would run backwards" maybe less controversial ( I know nothing in our sub-luminal universe can travel FTL, but just IF. -or better said, nothing can cross the barrier from subluminal to superluminal-).

We may picture an FTL signal in a spacecetime diagram by drawing a line tilted at more than 45º from the vertical (with the convention that 45º represents the speed of light).
But as long as we use an angle between 45 and 90º, such a line is still going upwards in the diagram.
In the y coordinate used for time, it still appears as running forward.
It looks like the line should go downwards to represent "backwards in time"

 

[robphy]

We may picture an FTL signal in a spacecetime diagram by drawing a line tilted at more than 45º from the vertical (with the convention that 45º represents the speed of light).
But as long as we use an angle between 45 and 90º, such a line is still going upwards in the diagram.
In the y coordinate used for time, it still appears as running forward.
It looks like the line should go downwards to represent "backwards in time"

Correct. But just realize that there is some observer that will (according to his reference frame) see the line going downwards. That is to say, there is a Lorentz Transformation that will transform that spacelike-ray to a spacelike ray with a negative time-component.

 

[Gerinski]

Thanks again, the lorenz transformation was the missing "trick" to make a line at 50º (up) look as going backwards in time (down) from the sun's frame of reference.

Sorry one more (maybe absurd?) question:

IF light from alpha-centauri would arrive to us in just 2 years (our time) after having been emitted instead of the usual 4 years, (i.e. IF light from alpha-centauri came to us FTL), would we notice anything abnormal in it ? would it really convey an image of how our neighbour star will look in the future ?