Show that if 0<v1<c and 0<v2<c are two velocities pointing in the same direction, the relativist

Show that if 0<v1<c and 0<v2<c are two velocities pointing...

Key Concepts

Relativistic Velocity Addition

In special relativity, the addition of velocities does not follow the simple arithmetic (Galilean) rule because the speed of light acts as an invariant upper limit. The relativistic velocity addition formula, v = (v? + v?) / (1 + (v?v?/c²)), ensures that whenever you combine two subluminal velocities (velocities less than the speed of light), the resulting velocity is also less than c. This non-linear formula reflects the fact that time and space are intertwined in a way that prevents any object from exceeding the speed of light.

Speed of Light as an Upper Limit

One of the cornerstones of special relativity is that the speed of light, c, is the same in all inertial frames and represents an absolute speed limit for any object with mass. This constraint forces the nature of velocity addition to accommodate the fact that even when individual velocities are high (e.g., greater than 0.5c), their relativistic combination can never reach or exceed c. This intrinsic ceiling on speed is critical for maintaining causality within the framework of relativity.

Nonlinear Composition and Its Implications

The nonlinear nature of the relativistic addition of velocities means that the combined effect of two velocities is not simply their arithmetic sum. Instead, the denominator (1 + (v?v?/c²)) serves to moderate the overall result, ensuring that while the composite velocity is always greater than each individual velocity (under the conditions specified), it remains bounded below c. This property is essential in all relativistic systems to prevent any scenario where the combined velocity could inadvertently exceed the universal speed limit.

Transcript

00:01 Hello, and in this question here, we're going to show that the maths of special relativity is consistent with the fact that you can't travel faster than speed of life.

00:09 To do this, we're going to consider an initial coordinate system at rest, which is s.

00:16 The vertical axis is time and the horizontal axis is position.

00:22 We have a second coordinate system, s prime, where s prime moves at a relative velocity v1 relative to s.

00:32 The vertical axis is t prime and the horizontal axis is x prime.

00:40 For an observer in s prime, there is an object and this object is traveling with the velocity v2.

00:50 So this velocity v2 is the velocity of the object relative to an observer in s prime.

00:58 We want to find out what the velocity of this object v2 is relative to an observer.

01:06 Classically we would just expect that you would add the velocity of s relative to s prime and then the velocity of the object relative to s prime and that would be the velocity of the object to an observer in s however this equation would allow you to travel faster than the speed of light so this cannot be true, so we use special relativity instead.

01:36 So to derive a formula for what the object travelling, for the velocity of the object in s, we will introduce the lorentz transforms between the two coordinate frames as x prime equals gamma multiplied by x minus v1 multiplied by where v1 is the relative velocity between s and s prime and t prime is equal to gamma multiplied by t minus v1 times x over c squared where c is the speed of light and gamma is equal to the square root of one divided by v1 squared over c squared so we will now introduce infinitesimal versions of these lorentz transforms and say that dx prime is is equal to gamma dx minus v1 d t and d t prime is equal to gamma d t minus v1 d x over c squared so we have two equations equation one and equation two let's divide equation one by equation two we get d x prime divided by d t prime our gmas will cancel dx minus v1 d t all divided by d t minus v1 d x over c squared okay so what is d x prime divided by d t prime well we would say d x d t is the velocity of the object in x and this here this velocity which we're going to call v.

03:38 This here is going to be the velocity that we want to show is less than the speed of life.

03:45 This velocity here, in analogy, we will say that this is the velocity of the object.

03:56 So of the same object, this velocity here, this one here, is the velocity of the same object as this one here.

04:07 However, the crucial distance that this will be the velocity relative to an observer.

04:11 In s prime.

04:14 I've got a color there, but in s prime.

04:20 So what is the velocity of the object relative to the observer in s prime? well, if we scroll back up here in our diagram, we see here that that velocity is v2, which equals v2...

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