Line of Simultaneity

Line of Simultaneity is basically a line of events which happen at the same time. If we have a frame that time t is the vertical axis and length x is the horizontal axis, so the line of simultaneity is just a horizontal line that parallel to the x axis. Because points on the line all have same height t, they all happen at the same time.

Let’s say there is a two headed gun can fire 2 photons, one to the right the other one to the left at the same time. Photon speed is \(c\) the light speed. Superman’s speed is \(c/2\). To start, the gun and Superman are placed at the origin. When the gun fires the two photons, at the same time Superman chases the right photon with speed \(c/2\). We denote the right photon A, Superman B, two headed gun C, left photon D. On the 1st second mark, we can observe 4 events：

A the right photon traveled 300,000km, in 1 second.

B Superman traveled 150,000km, in 1 second.

C the gun did not move, in 1 second. (its own reference)

D the left photon traveled -300,000km, in 1 second.

When we plot these 4 events, we can see they are on a horizontal line parallel to the x-axis.

#Lorentz Transforms
x.prime <- function(x,v,t){ 
 (x-v*t)/sqrt(1-v^2/c0^2) 
}

t.prime <- function(x,v,t){ 
 (t-v*x/c0^2)/sqrt(1-v^2/c0^2) 
}

c0=300000
v=c0/2
t=1

A=c(c0*t,c0*t)
B=c(v*t,c0*t)
C=c(0,  c0*t)
D=c(-c0*t,c0*t)

plot(rbind(A,B,C,D))

Now let’s convert these 4 events into Superman’s frame by using Lorentz transforms. We can see the line becomes tilted, the height of the 4 points are all different, means the 4 events are no longer simultaneous.

A’ the right photon only traveled 173205.1km in 0.5773503 second.

B’ Superman did not move, time passed 0.8660254 second (his own reference)

C’ the gun traveled -173205.1km in 1.154701 second

D’ the left photon traveled -519615.2km in 1.732051 second

A2=c(x.prime(c0*t,v,t),c0*t.prime(c0*t,v,t))
B2=c(x.prime(v*t,v,t), c0*t.prime(v*t,v,t))
C2=c(x.prime(0,v,t),   c0*t.prime(0,v,t))
D2=c(x.prime(-c0*t,v,t),c0*t.prime(-c0*t,v,t))

plot(rbind(A2,B2,C2,D2),ylim=c(0,520000))

Note: Simultaneous lines of gun’s frame must be parallel to the gun’s frame x-axis. Simultaneous lines of superman’s frame must be parallel to the superman frame x-axis. We can draw simultaneous lines to intersect with t-axis to read times.

Unlike we did here, we do not need to plot spacetime graphs separately. As long we compute the tilting angles, we can plot frames into one single spacetime graph. The simultaneous lines of a frame would tilt an angle same as the x-axis tilts.

Sending signals beyond the light speed

When we have two entangled qubits, we spin the first qubit the second qubit would spin simultaneously. So people say we could build communications faster than light speed, called instantaneous communication. But according to relativity theory, when things move faster than light speed, an effect could happen before its cause. We can never have an effect happening before its cause. So here is a contradiction.

Let’s say we shoot an arrow, the arrow must leaves bow before hitting bullseye. If speed of the arrow is faster than the light speed, we may see, by drawing simultaneous lines in some frames of reference, the arrow hits the bullseye before leaving the bow. So the arrow cannot fly faster than the light speed.

And when everything moves slower than the light speed, the chronological order is conserved in all frames of reference.

Because photons are too slow. We place a bow 1 light second, 300,000km, away from the origin on the right. It shoots an arrow on the 4th second mark. And the arrow arrives at 6 light seconds, \(6c\), away from the origin, after 2 seconds. So the arrow travels 5 light seconds in total distance in 2 seconds, its speed is \(2.5c\). Superman travels at speed \(0.5c\). Let’s use Lorentz transforms to see what happens in the superman’s frame. We can see the bow shoots the arrow at 4.041452 second, and the arrow hits target at 3.464102 second. So it hits target before leaves the bow.

c0=300000
v=c0/2
t1=4
t2=6

event1=t.prime(c0,v,t1)
event2=t.prime(6*c0,v,t2)
event1#4.041452
## [1] 4.041452
event2#3.464102
## [1] 3.464102