Previous page : Invariance of Space-time intervals
These are diagrams describing positions in space-time, as mentioned previously. Usually, only one space dimension and the time dimension are used. These diagrams are known as Minkowski diagrams. Let us start with a diagram that one could draw in classical physics. We chose the x-axis as the space dimension and the y-axis as the space dimension. We also scaled the time axis so that a light beam from the origin tilted at 45 °. We will have ct on the vertical axis and x on the horizontal axis, or t on the vertical and x/c on the horizontal.
The red lines in the diagrams correspond to light beams. In the first diagram, we have an observer that we regard as stationary. Light travels the distance ct at time t, and because our scale on the time axis is ct, the red line will have a slope of one.
In the second diagram, we added a blue worldline corresponding to an observer moving to the right. If the observer sees itself as stationary, there must be a corresponding time axis following that worldline. If the observer is not moving, the time axis remains at the origin.
The slope of the red line will now be less than for the original observer if we count the slope relative to the time axis. The speed of light of that beam is less than c.
How to fix that? We can tilt the x-axis (the now axis) so that the distances from a point on the red line are equal to the time and x-axis.
In the diagram, we tilted the green x´-axis as much upwards as the blue t´-axis to the left. The dotted lines, following the skewed coordinate system, will now be equally long, and the red line will thus have a slope of one in the skewed and also in the stationary systems. Thus, the speed of light will be the same in both systems.
As the x-axis shows all the points where t=0, that is, the now for the observer in S, the x´-axis shows all the points of now for the observer in S.’
The points of that axis can be calculated using the Lorentz transform
On the x´-axis t´=0. This gives us
On the page linked below, you may examine a space-time diagram a bit more. On that page, which looks like the figure below, you can see the position of the green point, both in the coordinate system that is seen as stationary and also in the primed, moving coordinate system. You can enter the speed of the object as a fraction of c. Clicking the Enter button or the Enter key on your keyboard enters the value.
You may also change the speed by grabbing and moving the blue dot on the top of the diagram. You can use a mouse, a touchpad, or a touchscreen for this.
You can do the same with the green point.
The blue line is the primed time axis, the green line the primed space axis, and the red lines are the paths of light moving to or from the origin.
The lines from the green point show the coordinates in both systems. 
- Set the velocity to 0.5c.
- Move the green point to the coordinates (3, 6). What will the primed coordinates be?
- Now move the point to (3, 6). What will the primed coordinates be?
- You see the Andromeda galaxy on the horizon. Now you start to walk towards it at 5.0 m/s. How much would that change the “now” at Andromeda? The Andromeda galaxy is 2537000 ly away.
By the way, if we could transport ourselves instantaneously from one point to another, we could also travel in time, both forwards and backwards. Say we move forward at 5.0 m/s, then travel to the now in Andromeda. Then you start to travel toward the Earth at 5.0 m/s, then you jump to the Earth’s now. You will now be over 30 hours in the future. If you start your travels in the opposite direction, you will end up over 30 hours back in time.
The question is then, what do they mean by “instantaneous travel” in some science fiction?
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