This lesson and the next two are beyond the Class 12 syllabus. They are here because the photon's momentum, the energy \(E = mc^2\) and the energy of the nucleus all rest on them. Read them for understanding; lessons 1 and 2 are the syllabus lessons of this chapter.
Frames in which the first law holds
The chapters on relative velocity described one motion from two frames, the ground and a moving train. The chapter on accelerated objects added pseudo forces, which appear only in accelerating frames. A frame in which Newton's first law holds is an inertial frame. In it, a body with no force on it stays at rest or keeps a constant velocity. Any frame moving at constant velocity relative to an inertial frame is itself inertial.
The Rajdhani Express running at a steady \(30\,\text{m/s}\) on a straight track is, very nearly, an inertial frame. A car that is braking is not: a loose bottle on its seat slides forward with no force pushing it. A merry-go-round is not either, because every point on it accelerates towards the centre.
Nothing inside tells you that you move
Galileo noticed in 1632 that below the deck of a smoothly sailing ship, everything behaves as on land. Drops fall straight down, fish swim equally in every direction, and a ball thrown to a friend needs no extra effort one way. The same is true in a Rajdhani coach on a smooth straight stretch. A cup of tea stands still on the table, and a dropped coin falls straight to your feet. Without looking out of the window, you cannot tell whether the train moves or stands at the station.
In 1905 Albert Einstein raised this into a principle for all of physics, not only mechanics. Light, electricity and magnetism must obey it too.
The laws of physics are the same in every inertial frame. No experiment done inside a closed box can tell whether the box is at rest or moving at a constant velocity.
Which of these frames are inertial, to a good approximation?
Speeds that add, and one that does not
In the chapter on relative velocity, speeds simply added. A passenger throws a ball forward at \(10\,\text{m/s}\) in a train moving at \(20\,\text{m/s}\). A person on the platform sees the ball move at \(20 + 10 = 30\,\text{m/s}\). Now let the passenger switch on a lamp that shines forward. The same rule would give the platform observer a speed of \(c + 20\,\text{m/s}\) for the light.
Light is an electromagnetic wave, and the laws of electricity and magnetism fix its speed in vacuum at \(c = 3 \times 10^8\,\text{m/s}\). If the speed of light depended on the frame, those laws would differ from frame to frame. The first postulate forbids that. So physicists of the nineteenth century expected light to move at \(c\) only relative to one special medium, called the ether. An observer moving through the ether should find light faster one way than the other.
In 1887 Albert Michelson and Edward Morley tested this with the instrument of Fig 3.1. The Earth moves round the Sun at \(30\,\text{km/s}\). So light along one path should take a slightly different time from light along the other. The fringes should then shift as the instrument is turned. The instrument could detect a shift far smaller than the one expected. They found no shift at all.
The speed of light in vacuum has the same value \(c\) for every observer in every inertial frame. It does not depend on the motion of the source or of the observer.
So the passenger and the person on the platform both measure the lamp's light at \(c\), not \(c + 20\,\text{m/s}\). The rule "speeds add" works well for the ball and fails for light. We never notice the failure because the corrections go as \(v^2/c^2\). Even for an aircraft at \(300\,\text{m/s}\), \(v^2/c^2 = 10^{-12}\). For the ball in the train, the exact answer differs from \(30\,\text{m/s}\) by less than \(10^{-13}\,\text{m/s}\).
A train moves at speed \(v\). A lamp at its rear shines forward. What speed does a person standing on the platform measure for this light?
A passenger throws a ball forward at \(10\,\text{m/s}\) relative to a train that moves at \(20\,\text{m/s}\). What speed does a person on the platform measure for the ball, in m/s?
Two flashes and two observers
The second postulate has a strange consequence, and a train shows it best. Each place and instant at which something happens is an event. A train \(240\,\text{m}\) long passes a platform at half the speed of light. Passenger A sits at the middle of the train. Observer B stands on the platform. Just as A passes B, two flashes go off together at the two ends of the train, as the platform measures it.
The flashes are simultaneous for B on the platform. Passenger A sits at the middle of the moving train. Which flash reaches A first?
Press play in the figure above. Both pulses travel at \(300\,\text{m}\) per microsecond, and each has \(120\,\text{m}\) to go to reach B. They arrive together at \(0.40\,\text{μs}\), and B rightly says the flashes were simultaneous. A moves forward at \(150\,\text{m}\) per microsecond. A meets the front pulse at \(0.27\,\text{μs}\) and the back pulse only at \(0.80\,\text{μs}\). Choose the chip with the train at rest, and both pulses reach A and B together.
Now reason as A does. A sits at the middle of the train. Each flash left an end of the train and crossed half the train to reach A. By the second postulate, both pulses moved at \(c\) in A's frame as well. Equal distances at equal speeds would take equal times. Yet the front pulse arrived first. So in A's frame the front flash happened first.
Neither observer is wrong. Each uses the same laws and the same speed of light, and each reasons correctly in their own frame. Whether two events at different places happen at the same time depends on the frame.
Two events at different places that are simultaneous in one inertial frame are not simultaneous in another inertial frame moving along the line joining them. The event at the front, in the direction of that frame's motion, happens first for it.
The front flash is not "really" first, and B is not fooled by the travel time of light. B allowed for the travel time: B stands midway and both pulses cross equal distances. A allowed for it too. Both calculations are right, and they give different answers.
A train \(300\,\text{m}\) long moves at half the speed of light past a platform. Flashes go off together at its two ends, as measured on the platform, just as its middle passes an observer there. Take \(c = 300\,\text{m/μs}\). In the platform frame, after how many microseconds does the passenger at the middle of the train meet the light from the front?
State the two postulates of special relativity in your own words, and say which one decides that A and B disagree about the flashes.
First, the laws of physics are the same in every inertial frame, so no experiment in a closed box can detect its uniform motion. Second, the speed of light in vacuum is c for every inertial observer, whatever the motion of the source. The second postulate decides the disagreement: A must take both pulses to move at c in the train's frame, so the pulse that arrives first must have left first.
In Cleveland, Albert Michelson and Edward Morley floated their interferometer on a bath of mercury, so that it could turn smoothly. Mirrors sent each beam to and fro several times, making each path about eleven metres long. They expected the fringes to shift by about four tenths of a fringe as the instrument turned. They could have seen a shift of one hundredth. They found none, at any time of day or season. Eighteen years later, Einstein's second postulate made the null result exactly what one should expect.
- Adding the speed of the source to the speed of light.The speed of light in vacuum is \(c\) in every inertial frame, whatever the source does.
- Thinking the relativity of simultaneity is an illusion of light delay.Each observer corrects for the travel time. The disagreement remains after the correction.
- Treating a braking or turning vehicle as an inertial frame.Only a frame at rest or moving at constant velocity relative to an inertial frame is inertial.