Is observed speed of an object coming towards you higher than the real speed, even >c?

A spaceship $S$ takes off from planet $B$ at $v=\frac{c}{2}$ towards planet $A$ , $10 ly$ away. $20$ years later it lands on planet $A$ .

When an observer $O$ on planet $A$ sees spaceship taking off , the light from the take-off has been travelling for $10$ years and the spaceship is already half-way to planet $A$ .

A              <- 10 ly ->           B
O                S(real)              S(observed)


When the spaceship lands another $10$ years later the observer on planet $A$ sees the time from the observed take-off to landing to be $10$ years. So the observer thinks the spaceship has been travelling at the speed of light but the real speed is $\frac{c}{2}$ .

I have read other articles but I can't understand what I am missing here.

Best Regards, Lars Ekman

• haven't you answered your own question? – Alex Robinson Apr 7 '18 at 10:24
• So speeds >c can be observed? – lgekman Apr 7 '18 at 10:25
• related – lgekman Apr 8 '18 at 12:00

Yes apparent speeds $>c$ can be "observed". But this does not mean that the spaceship has actually travelled faster than $c$. The observer must make allowance for the finite speed of light, which gives him the information about the position of S at different times.

In your example the spaceship S arrives 10 years after the the light which informs the observer that it has taken-off from a planet which is 10 LY away. So the journey time for S is actually 20 years not 10.

Think of S and the light from the launch as being in a race to planet A. They start at the same time. The light reaches A first, after 10 years. S reaches A 10 years after the light - ie 20 years after the start of the race. So the journey takes S 20 years to complete.

• Thanks. I didn't think speeds >c could be observed, not even apparent speeds. – lgekman Apr 7 '18 at 11:10
• – sammy gerbil Apr 7 '18 at 11:13
• In that example nothing moves (as pointed out). Maybe you can say the point-of-impact of the photones moves at >c but that is actually new points-of-impact all the time. – lgekman Apr 8 '18 at 7:05

In relativity, there is no absolute position or time. Each observer simply assigns to each point in spacetime x,y,z, and t coordinates. This is a "coordinate system". One can choose whatever coordinates system one likes, but some are more useful than others, and some are more natural than others. If A and B are 10ly away from each other, it can be observed that it takes light 20 years to travel from A to B and back again. There is no way to directly observe how long it takes light to travel from B to A. If one asserts that light travels instantaneously from B to A, but takes 20 years to go the other way, there is a coordinate system in which that is true. However, as there is no reason to think that the universe prefers certain directions, it is natural to assume that it takes light 10 years each way. A coordinates system consistent with this assumption would assign the event of light leaving B a time coordinate 10 years earlier than the event of that light arriving at A. That is:

(light leaves B)t = (light arrives at A)t - 10.

According to this coordinate system, a person calculating the speed of the ship would take the distance traveled divided by time taken, and to calculate time taken, they would take the difference in time coordinates of the two events. That is, they would take the time coordinate of the ship arriving at A, and subtract the time coordinate of it leaving B. Note that the last quantity is the time coordinate of the ship leaving B, not the time coordinate of the person on A seeing the ship leaving B. The time of an event is the time it occurs, not the time that you see it.

In relativity, the "observed" time of an event is the time coordinate that one's coordinate system assigns to that event, not the time that it is seen.

• "Apparent speed" is defined in terms of the arrival time of light from the events. It's not a useful quantity for talking about theory (which is, indeed done in terms of coordinate time) but is the relevant quantity if you want to talk about responding to events involving relativistic velocities. – dmckee --- ex-moderator kitten Apr 13 '18 at 19:03