Split from this thread.
Causality is determined by the light cone. Events inside the light cone are causally related to its centre. Events outside the cone are not. The boundaries of the cone are the worldlines of a light ray. If you can break these boundaries (travel faster than light) you can break causality.
The spacetime interval is defined as s2 = x2 - c2t2. s2 is the same for all inertial observers. If s2 < 0, the temporal separation is greater than the spatial separation, so the events can be causally related. That is, if tA < tB, then A can be the cause of B. If I shoot you, moments later you die. This happens in this order for every observer, no one sees you dying before I shoot. Causality is not relative.
If tachyons (particles travelling faster than light) existed, this would change. Check, for example, the section on tachyons here.
Another way of looking at this is noticing that a particle travelling faster than light in one reference frame R can be travelling backwards in time in another reference frame R'. In other words, FTL travel = time travel. And it is obvious that time travel implies several paradoxes and contradictions. To see this, consider the following figure:
The axes in black (red) represent R (R'). R' is a reference frame moving with some velocity parallel to the x axis relative to R, so its axes appear slanted. The grey line is the worldline of a light ray. Notice how it bisects both frames. The green line represents the worldline of a tachyon. As you can see, it takes less time for the tachyon (in R) to cover the same distance as the light ray. Simultaneous events in R are those living on the same horizontal line. Simultaneous events in R' are those living on a line parallel to the axis x', such as the dotted red line.
Consider now that the origin is event A, with coordinates (0,0) in both systems. The point of intersection between the green line and the dotted line is event B. This event has positive time in R, but negative time in R'. So the tachyon is travelling back in time in R'.
And now for something completely different:
This is different. Doppler shift does not mean that photons change their frequency during their travel, it just means that different observers will measure different frequencies. With the oscillation of neutrinos, the same particle that starts its voyage as an electronic neutrino may arrive at Earth as a tau neutrino. The same observer sees the same particle change.
Causality is determined by the light cone. Events inside the light cone are causally related to its centre. Events outside the cone are not. The boundaries of the cone are the worldlines of a light ray. If you can break these boundaries (travel faster than light) you can break causality.
The spacetime interval is defined as s2 = x2 - c2t2. s2 is the same for all inertial observers. If s2 < 0, the temporal separation is greater than the spatial separation, so the events can be causally related. That is, if tA < tB, then A can be the cause of B. If I shoot you, moments later you die. This happens in this order for every observer, no one sees you dying before I shoot. Causality is not relative.
If tachyons (particles travelling faster than light) existed, this would change. Check, for example, the section on tachyons here.
Another way of looking at this is noticing that a particle travelling faster than light in one reference frame R can be travelling backwards in time in another reference frame R'. In other words, FTL travel = time travel. And it is obvious that time travel implies several paradoxes and contradictions. To see this, consider the following figure:
The axes in black (red) represent R (R'). R' is a reference frame moving with some velocity parallel to the x axis relative to R, so its axes appear slanted. The grey line is the worldline of a light ray. Notice how it bisects both frames. The green line represents the worldline of a tachyon. As you can see, it takes less time for the tachyon (in R) to cover the same distance as the light ray. Simultaneous events in R are those living on the same horizontal line. Simultaneous events in R' are those living on a line parallel to the axis x', such as the dotted red line.
Consider now that the origin is event A, with coordinates (0,0) in both systems. The point of intersection between the green line and the dotted line is event B. This event has positive time in R, but negative time in R'. So the tachyon is travelling back in time in R'.
And now for something completely different:
This is different. Doppler shift does not mean that photons change their frequency during their travel, it just means that different observers will measure different frequencies. With the oscillation of neutrinos, the same particle that starts its voyage as an electronic neutrino may arrive at Earth as a tau neutrino. The same observer sees the same particle change.
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). I've listened to a lot of the stuff from The Teaching Company and seen a lot of PBS shows and stuff. And while they have done a great deal for me in regard to the reference frame stuff, they either didn't properly(or it has just slipped from my mind since then) explain why FTL would be time travel.