Special Relativity

Special Relativity - Chapter 1 of Modern Physics, 3rd ed. by Serway, Moses, and Moyer

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• Created by: Victoria
• Created on: 23-04-12 16:36

Postulates of Special Relativity

1. The laws of physics are the same in all reference systems that move uniformly with respect to one another.

2. The speed of light in a vacuum, 3x10^8 m/s, is independent of the motion of the observer or the source of the light.

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Galilean transformation of cooordinates

x' = x - vt
y' = y
z' = z
t' = t

Note: the fourth coordinate, time, is assumed to be the same in both inertial frames - in classical mechanics, all clocks run at the same rate regardless of their velocity, so that the time at which an even toccurs for an observer in S is the same as the time for the same event in S'.

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u'_x = u_x - v

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Example 1.1

F = ma has been shown to hold by an observer in inertial frame S. Show it also holds for an observer in S' or is covariant under the Galilean transformation.

With Galilean transformations:

a = a'
m = m'

So F = m'a'.

If F depends only on the relative positions of m and of the particles interacting with m, then the change in x is invariant and F = F'.

So F' = m'a'.

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Michelson-Morley

• Performed in 1887 by Albert A Michelson and Edward W Morley
• Was designed to detect small changes in the speed of light with motion of an observer through the ether
• Failed, contradicting the ether hypothesis
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Overview of consequences of Special Relativity

• The distance between two points and the time interval between two events depend on the frame of reference in which they are measured
• Events at different locations that occur simultaneously in one frame are not simultaneous in another frame moving uniformly past the first

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Simultaneity

• According to Einstein, a time interval measurement depends on the reference frame in which the measurement is made.
• Two evens that are simultaneous in one frame are, in general, not simutaneous in a second frame moving with respect to the first.
• Simultaneity is not an absolute concept, but one that depends on the state of motion of the observer.
• The principle of relativity states that there is no preferred inertial frame of reference.
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Time dilation

delta t = delta t' / sqrt [1 - (v^2/c^2)] = gamma*delta t'

Delta t' is proper time
In general, proper time, delta t_p, is defined as the time interval between two events as measured by an observer who sees the events occur at the same point in space
Proper time is always the time measured by an observer moving along with the clock

delta t = gamma delta t_p

A moving clock runs slower than a clock at rest by a factor of gamma

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Example 1.2

The period, T, of a pendulum is 3.0 s in the rest frame of the pendulum. What is T when measured by an observer moving at a speed of 0.95c with respect to the pendulum?

Proper time is 3.0 s.

T = gamma * T' = (1 /  sqrt [1 - (0.95)^2/c^2]) * 3.0 s = (3.2)(3.0 s) = 9.6 s

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Length contraction

The proper length, L_p, of an object is defined as the length of the object measured by someone who is at rest wiht respect to the object

L = L_p (1 - v^2/c^2)^1/2

If an object has a proper length L_p when measured by an observer at rest with respect ot the object, when it moves with speed v in a direction parallel to its length, its length L is measured to be shorter according to L = L_p (1 - v^2/c^2)^1/2

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Example 1.3

A spaceship is measured to be 100 m long while at rest with respect to an observer. If it flies by the observer with a speed of 0.99c, what length will the observer find for the spaceship?

Proper length is 100 m.

L = L_p (1 - v^2/c^2)^1/2 = (100 m) [1 - (0.99c)^2/c^2]^1/2 = 14 m

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Lorentz transformations

• Lorentz coordinate transformation is a set of formulas that relates the space and time coordnates of two inertial obervers moving with a relative speed v
• To obtain the inverse Lorentz transformation of any quantity, simply interchange primed and unprimed variables and reverse the sign of the frame velocity

S -> S'

x' = gamma (x - vt)
y' = y
z' = z
t' = gamma (t - vx/c^2)

where gamma = [1 - (v^2)/(c^2)]^-1/2

S' -> S

x = gamma (x' + vt')
y = y'
z = z'
t = gamma (t' + vx'/c^2)

where gamma = [1 - (v^2)/(c^2)]^-1/2

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Lorentz velocity transformation

S -> S'

u' = u - v/1 - (uv/c^2)

S' -> S

u = u' + v/1 + (u'v/c^2)

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