Of the systems we ordinarily use to transmit information,...

Of the systems we ordinarily use to transmit information, the fastest ones - radio, television, phone conversations carried over fiber-optic cables - use light. Nevertheless, we might wonder whether it is possible to transmit information at speeds greater than $c$. The purpose of this problem is to show that if this was possible, then special relativity would have problems with catusality, the principle that the cause should come carlier in time than the effect. Suppose an event happens at position and time $x_1$ and $t_1$ which causes some result at $x_2$ and $t_2$. Show that if the distance berween $x_1$ and $x_2$ is greater than the distance light could cover in the time between $t_1$ and $t_2$, then there exists a frame of reference in which the event at $x_2$ and $t_2$ occurs before the one at $x_1$ and $t_1$.

Key Concepts

Spacetime Interval

The spacetime interval is an invariant quantity in special relativity that combines differences in both space and time between events. It categorizes separations into timelike, spacelike, or lightlike, which reflects whether a causal connection is possible. Spacelike intervals indicate that events cannot be causally related without violating the speed of light constraint.

Lorentz Transformation

The Lorentz transformation equations provide the mathematical mechanism to translate the space and time coordinates of events between different inertial frames. They reveal how time intervals and spatial separations mix under relative motion, leading to effects such as time dilation and length contraction, and play a crucial role in analyzing the relative order of events.

Light Cone

The light cone is a geometric representation in spacetime that delineates the boundary of causal influence. It partitions events into those that can potentially affect or be affected by a given event (inside the light cone) and those that are causally disconnected (outside the light cone), emphasizing the fundamental speed limit set by the speed of light.

Special Relativity

Special relativity is the framework developed by Einstein that establishes the constancy of the speed of light in all inertial frames and the equivalence of physical laws in those frames. This theory fundamentally alters our understanding of space and time, showing that measurements of distances and time intervals can differ between observers moving relative to each other.

Causality

Causality is the principle that a cause must precede its effect in time. In the context of special relativity, it implies that the ordering of events must be preserved in all inertial frames for events that are causally connected, ensuring that information or influence does not travel backwards in time.

Transcript

00:01 Here we're going to look at causality in special relativity.

00:06 Causality is the idea that an event in frame one causes another event in frame one.

00:16 And the worry is, will special relativity in particular, the lorentz transformation, ruin that idea? so this is a topic that is explored much in science fiction.

00:32 But in order to show that coralsity is preserved, we're going to need the lorentz transformation equations.

00:42 And we'll just stick in one dimension.

00:47 So we have the transformation on the position coordinates between frame, unprimed, and primed.

00:58 As well as the time coordinate between primed and unprimed.

01:08 And to start off this, we'll take a look at a simple case to see how this goes.

01:15 Let's have a case where we'll put unprimed frame.

01:24 We'll have two events occur at the same location.

01:28 And these may be on earth, we'll call earth the same location, which is x2 equals x1 for the two events.

01:40 And event one at time one is your mom is born.

01:53 Okay, so mom is born on earth.

02:00 And event two at time t2 is you are born.

02:13 So that is definitely a causal connection there.

02:18 So now let's see what those.

02:20 Times look like in terms of frame s prime.

02:25 So we'll put somebody in a rocket ship out there.

02:28 We don't know where they are, but we're most interested in the difference t2 prime minus t1 prime.

02:43 And for that we are going to use lorentz transformation on time.

02:47 We'll call these equations one and equations two.

02:52 So this is fairly easy.

02:54 To put in for equation two, basically the positions x are the same.

03:02 So we could write all that down, t2 prime in terms of t2, and take the difference.

03:25 And of course, the position information will go away.

03:29 But what we see very readily is that the way that those two times are organized is still t2 comes after t1.

03:45 In the earth frame means t2 prime comes after t1 in the moving frame of this rocket ship.

03:54 Let's just make it a rocket ship.

03:57 And the reason why we can say that is we know that gamma is a positive quantity, just a constant that's greater than one.

04:08 So the time order is perfectly valid, although the gamma factor shows time dilation.

04:21 Okay, so let's take a look at a slightly different case.

04:26 Let's kind of turn this around and make two events happen in frame s...

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