Special Relativity, Light cones in Minkowski Diagrams

[king vitamin]

Homework Statement

Show that a point Q outside the "light cone" can occur either before or after a point P inside the light cone (eg: at the origin), depending upon the frame of reference. Show that a point R inside the light cone is always in the future or in the past with respect to P, irrespective of frame S'.

Homework Equations

this is just diagrams

The Attempt at a Solution

I know this isn't a difficult problem, but I'm having trouble conceptualizing it because I've never even seen light cones displaced from the origin, let alone in other frames of reference. How does a light cone transform between reference frames? I can't picture it because the x' and ct' axes are no longer orthogonal when drawn over S, so I'm not sure how to represent things.

 

[tiny-tim]

Hi king vitamin!

Forget the shape of the cone …

the question is just asking you to prove that x² > c²t², then in some frames t' > 0 but in others t' < 0 (while if x² < c²t², then in all frames, t' has the same sign).

 

[tomcue]

In Special Relativity, the concept of a light cone is essential for understanding the relationship between space and time. A light cone is a geometric representation of the paths that light can take in a given reference frame. It is a cone-like shape with the tip at the origin, representing the present moment, and the sides extending outwards to represent the past and future.

In Minkowski Diagrams, the x-axis represents space and the ct-axis represents time. The light cone is represented by a 45 degree line, with the tip at the origin and the sides extending outwards. Any event that occurs within the light cone is considered to be in the causal future or past of the origin event, depending on the direction of the event.

Now, in the given scenario, we are considering a point Q outside the light cone and a point P inside the light cone. This means that the event at Q is not causally connected to the event at P, as it is outside the light cone. This also means that the event at Q can occur at any time, either before or after the event at P, depending on the frame of reference. This is because the concept of simultaneity is relative in Special Relativity, and different frames of reference will have different perspectives on the timing of events.

On the other hand, for a point R inside the light cone, it will always be in the future or past with respect to P, irrespective of the frame of reference. This is because any event within the light cone is causally connected to the event at the origin, and therefore, will always be considered to be in the future or past of the origin event.

To visualize this, we can consider the transformation of the light cone between frames of reference. As you mentioned, the x' and ct' axes are no longer orthogonal when drawn over S, so we cannot simply rotate the light cone to represent the new frame. Instead, we have to use the Lorentz transformation equations to calculate the coordinates of the light cone in the new frame.

In conclusion, the position of a point outside the light cone is relative and can occur either before or after the point inside the light cone, depending on the frame of reference. However, for a point inside the light cone, it will always be in the future or past with respect to the origin event, regardless of the frame of reference. The transformation of the light cone between frames of reference can be calculated using the Lorentz transformation