Planes of Simultaneity on Minkowski diagrams As seen from a "stationary" reference frame, Why do lines of simultaneity for fast-moving observers run parallel to distance axis instead of perpendicular to the time axis. Obviously, in a regular t vs x diagram, both are true because the angle between the axes is 90 degrees, but for moving observers the angle between the two axes is not 90 degrees. I can't seen to conceptually understand why. Had I not looked it up, I would have assumed that lines of simultaneity run perpendicular  to the "slanted" t' axis.
Thanks!
 A: The plane of simultaneity in a particular observer's reference frame is defined to be the set of events which that observer views as simultaneous, i.e. the events that he observes to be "orthogonal to the time axis".* 
He does not care what you think is orthogonal to his time axis. His plane of simultaneity is defined in his reference frame and then maps to whatever plane it maps to in your reference frame.
*This is a horrible way to think about relativity, but we will go with it
A: On a t vs x diagram, the t'-axis is tilted at an angle of $\arctan(v/c)$ with respect to the t-axis. A line of simultaneity for the moving observer makes an angle of $\arctan(v/c)$ with the x-axis. As a result the line of simultaneity(also called the x' axis) makes an angle of $90^{\circ}-2*\arctan(v/c)$ with the t'-axis which is less than $90^{\circ}$. So the angle between the two axes is not $90^{\circ}$ but less. In order to prove that a line of simultaneity makes the said angle with the x-axis you can refer to the book Special Relativity by A.P. French. A little bit of geometry will of course be called for.
