Introduction to special relativity

Introduction to special relativity

By the end of this topic you should be able to:

•state the meaning of the term frame of reference;•state what Galilean relativity means;•solve problems of Galilean relativity;•understand the significance of the speed of light;•state the two postulates of the principle of relativity;•appreciate that absolute time does not exist, and that simultaneity is a relative concept.

Frames of referenceThe observer along with the rulers and clocks that he or she uses to measure distances and times constitute what is called a frame of reference. If the observer is not accelerated, the frame is called an inertial frame of reference.

Frames of referenceTo discuss motion we must first specify the frame of reference with respect to which the motion is to be described. Thus, an observer with rulers and clocks who is firmly attached to the surface of the earth will measure different things than an observer traveling in a train.

Absolute Time

Assumption: two observers always agree on what the time coordinates are; in other words, time is common to both observers.

“Absolute, true and mathematical time, of itself, and from its own nature, flows equably without any relation to anything external.”

Relativeness of motionIt is impossible for one of the observers to claim that he or she is “really” at rest and that the other is “really” moving. There is no experiment that can be performed by the train observer, say, that will convince her that se “really” moves. Whatever results the train observer gets out of her experiments, the ground observer also gets out of the same experiments performed in his ground frame of reference.

Galilean transformations

x’ = x – vt

t’ = t

Galilean transformations are the relation between coordinates of events when one frame moves past the other with uniform velocity on a straight line.

Observers in both frames are equally justified in considering themselves to be at rest and the descriptions they give are equally valid.

Non-inertial frames

A non-inertial frame, by contrast, is an accelerating frame, and in this case it is possible to distinguish the observer who is “really” moving.

Law of addition of velocities

u = u’ + v

Example question

A ball rolls on the floor of a train at 2 m/s (with respect to the floor). The train moves with respect to the ground (a) to the right at 12 m/s, (b) to the left at 12 m/s. What is the velocity of the ball relative to the ground?

This apparently foolproof argument presents problems.

It is said that Einstein, as a boy, asked himself what would happen if he held a mirror in front of him and ran forward at the speed of light. With respect to the ground, the mirror would be moving at the speed of light. Rays of light leaving young Einstein’s face would also be moving at the speed of light relative to the ground. This meant that the rays would not be moving relative to the mirror, hence there should be no reflection in it. This seemed odd to Einstein. He expected that looking into the mirror would not reveal anything unusual.

At the end of the 19th century, considerable efforts were made to detect variations in the speed of light depending on the state of motion of the source of light. The experimental result was that no such variations were detected!

The speed of lightIn 1864, James C. Maxwell synthesize the laws of electricity and magnetism, and discovered that accelerated electric charges produced a pair of self-sustaining electric and magnetic fields at right angles to each other, which eventually decoupled from the charge and moved away from it at the speed of light.

One prediction of the Maxwell theory was that the speed of light is a universal constant.

The speed of light

This result is in conflict with Galilean relativity.

EtherMaxwell believed that electromagnetic waves requiered a medium (the ether) for their propagation. He was forced to admit that his equations were not in fact valid in all frames of reference but only in a small subset, namely those inertial frames that were at rest relative to the ether.

If while moving through the ether you were to emit a light signal in the direction of motion, the velocity of light was expected to be less than the velocity of light would be if you were at rest relative to the ether.

Special relativityEinstein threw away the entire notion of the ether. Electromagnetic waves do not requiere a medium and the speed of light in a vacuum is the same for all observers. The laws of physics are the same in all inertial frames of reference. This means immediately that the laws of Galilean relativity have to be modified. Einstein was thus led to a modification of the transformation laws from

to

Postulates of theory of special relativity

•The laws of physics are the same in all inertial frames.

•The speed of light in a vacuum is the same for all inertial observers.

Consequence: Absolute time does not exist. Observers in motion relative to each other measure time differently!!! Space and time are now inevitably linked and are not independent of each other as they were in Newtonian mechanics.

Relativity of simultaneity

Events that are simultaneous for one observer and which take place at different points in space, are not simultaneous for another observer in motion relative to the first.

On the other hand, if two events are simultaneous for one observer and take place at the same point in space, they are simultaneous for all other observers as well.

Example questionObserver T is in the middle of a train carriage that is moving with constant speed to the right with respect to the train station. Two light signals are emitted at the same time as far as the observer, T, in the train is concerned.a) Are the emissions simultaneous for observer G on the ground?b) The signals arrive at T at the same time as far as T is concerned.

Do they arrive at T at the same time as far as G is concerned? According to G, which signal is emitted first?

Exercises