1) Special Relativity. Statement: Imagine this situation: Alice stands in New York City while Bob, aboard a plane departing from Boston, directly crosses over Alice at t=0. Disregard the vertical distance between them. Alice is linked to Frame A, termed the "fixed frame," while Bob is associated with Frame B, designated as the "moving frame." Let's examine two space-time events: 1) The moment when Bob's plane is precisely above Alice serves as the space-time origin for both frames of reference. 2) A luminous beacon is illuminated at coordinates xo and time to in Alice's frame, making her first witness the flash at time to + xo(c = 1). Envision this beacon as situated several kilometers east of New York City. In the fixed frame, the light flash is thus considered causal. In this problem, we explore causality within the idealization of classical physics. The assurance of Lorentz invariance implies that events causally connected in one frame maintain that causal relationship in any other inertial frame. Your objective is to visually demonstrate and validate this consistency. Question: In classical physics, events can only be causally connected (lie on the same world line) if they fall within each other's light cone. In this scenario, where only a single relevant spatial dimension exists, the space-time (M) is 2-D. Nevertheless, considering that t = 0 is an arbitrary choice, let's encompass the entirety of (M) and not limit ourselves to positive times. Create a 2-D space-time sketch, viewing it from the fixed frame's perspective, illustrating regions of time-like and space-like separation for Events 1 and 2. Clearly indicate the regions where Event 2 is causally connected to Event 1, keeping in mind that Event 1 is positioned at the origin.