Physics Project : Relativity

Basic Ideas of Relativity

Mrinal Mohit MR Ganesh Palash Utkarsh Sharma

Introduction Everything is relative. There is no possible way by which one can say that there exists a state of absolute rest or motion. But here arises a problem...how do we measure speed???? In relativity, the measure of speed is done relative to the speed of light. This is because relativity makes itself noticeable….a lot more noticeable at some appreciable fraction of the speed of light. The entire idea of relativity was given by Albert Einstein not when he was working in a lab but on paper while imagining himself to be traveling at speed of light... At such high speed weird things start happening…like time slows down…things appear stationary…and you can’t see the back of the machine you are traveling in…matter gets squashed…something that we have never experienced…but the reason behind them is logically sound, so perfect that it is bound to happen if ever such speeds could be attained.

Intrigued?? Well then take a plunge in the deep waters of Relativity…

Principle of Relativity The Principle of Relativity states that the laws of nature are the same whether the observer is standing still or moving in a straight line with constant velocity. This seemingly obvious statement implies that all experiments will produce the same results irrespective of the experimenter’s state of rest or uniform motion with respect to a stationary reference frame. E.g. experiments conducted in a train in motion (with constant velocity) will produce same results as those performed on ground. However, this principle is not applicable if the train is accelerating. The Principle also implies that since no measurement will be affected by uniform motion, there is no measurement you can make that will tell you whether you are stationary or moving without acceleration. This is in fact implied even by Newtonian concepts of relative motion.

Time Dilation Time dilation is a phenomenon described by the theory of relativity which causes observers moving with different velocities to observe errors in each other’s clocks. Imagine yourself to be in a train, which is (somehow) travelling at the speed of light. You turn around and look towards the back of the train. Spookily, you can’t see it! One can only see an object when light reflected from the object enters one’s eyes. But since, you yourself are travelling at the speed of light; the light rays from the back of the train can never reach you. This appears to be a foolproof test which could that you are moving (at the speed of light). But this contradicts the Principle of Relativity, which is a very basic law. Let’s assume that the Principle is right and then analyze the situation. This means that the carriage’s back must look normal to you, irrespective of your velocity. This further implies that the light from the back of the carriage still reaches your eyes as it normally would. This can only happen if light ignores the motion of the train entirely and travels at an apparent absolute speed. i.e. The speed of light is same for everyone, if they are not accelerating. For example, our everyday common sense tells us that if while standing on a moving cart we hit a golf ball off the front, the speed of the cart adds to the speed of the golf ball. But If we were to shine a flashlight in the same direction as we had hit the ball we would find that the speed of the cart does not add to the speed of the light beam. Its speed would be the same as if the cart was not moving at all.

Light Clock Experiment Imagine having two horizontal mirrors facing each other such that one mirror is spaced above the other by the distance d. Also imagine that there is a pulse of light that bounces vertically between the two mirrors as shown in the left part of the drawing below.

Suppose our "light clock" was traveling sideways at a very high (but constant) speed. Now the pulse would follow the "saw tooth" path shown on the right side of the drawing. The light must travel a greater distance now to make a round trip. Since its speed is the same as before (remember, the speed of light is not changed by the speed of its source), it will take longer to make a round trip. So our "light clock" takes longer to count out its intervals. Another way of saying this is that the clock "ticks" more slowly. In other words, time itself slows down in a moving object. But note that time will appear to be normal to a person who’s travelling with the light clock. Note that if the speed of light were not constant (or if a “normal” thing was bouncing up and down), the horizontal speed of the clock would have vectorially added to the speed of the pulse. Then, the light clock would not have slowed down since the pulse's greater speed would have compensated for the longer distance of the "saw tooth" path.

Let’s assume that the light clock is travelling at half the speed of light i.e. at 0.5c ms -1.We want to compare how fast the clock ticks when it’s moving past with how fast it ticks when it isn’t. Then we’ll know exactly how speed affects time. Let’s say d = 1 meter. Light travels 3x108 meters per second. To travel one meter, it will take (1/3)x10-8 seconds, i.e. 3.3 nanoseconds When the clock is moving with a speed of 0.5c, the horizontal line will be half as long as the sloping line. i.e. the angle between the lines will be cos-1 0.5 = π/3 C

3.3 nanoseconds π/3 A

B

From trigonometry, we find that 𝐴𝐶 =

𝐴𝐵 3.3 𝜋 = 3 = 3.8 𝑛𝑎𝑛𝑜𝑠𝑒𝑐𝑜𝑛𝑑𝑠 sin 3 2

So that means, if it takes 3.3 nanoseconds for the clock to tick when it’s still, it takes 3.8 nanoseconds to tick when it’s moving past you at half the speed of light.

Albert Einstein derived an equation for the result, which is

𝑡=

Where,

𝑇 𝑣2 1− 2 𝑐

t = amount of time that passes for you, according to a clock you carry T= amount of time that passes on the moving object v = velocity of moving object c = speed of light

In the above example,

3.3

𝑡= 1−

150,000,0002

= 3.8

300,000,0002

Since time flows at different rates depending on how fast you’re moving, that meant that different people would see different clocks telling different times. i.e. there is no such thing as meanwhile.

Length Contraction Length contraction is the physical phenomenon of a decrease in length detected by an observer in objects that travel at any non-zero velocity relative to that observer. This is usually only noticeable, however, at a substantial fraction of the speed of light; and the contraction is only in the direction parallel to the direction in which the observed body is travelling. Let’s say that four friends A, B, C and D decide to measure the length of a train. They can do so easily while it is at rest (relative to them) but what if it’s travelling near the speed of light? We’ll assume that they’re armed with little ray-guns which emit beams of light. They’re also wearing accurate watches which initially show the same time.

A fires the gun which hits B after 40 nanoseconds (according to B’s watch). So according to them, the length of the train is 𝑙𝑒𝑛𝑔𝑡ℎ = 𝑠𝑝𝑒𝑒𝑑 × 𝑡𝑖𝑚𝑒 𝑙 = 3 × 108

× 4 × 10−8 = 12 𝑚

Now suppose the readings are taken by C and D, who are standing outside the train. C notes the time at which A fires the gun while D notes the time when B gets hit. According to D, by the time the ray hits B, the train too has moved forward, and therefore he notes the time to be lesser than 40 nanoseconds, let’s say 30 nanoseconds (remember that the train is travelling VERY fast) So 𝑙 = 3 × 108

× 3 × 10−8 = 9 𝑚

This implies that moving objects shrink relative to the reference frame. The equation for calculating this contraction, as given by Hendrik Lorentz is

𝐿′ Where,

𝑣2 =𝐿 1− 2 𝑐

L’ = proper length (the length of the object in its rest frame) L = length observed by an observer in relative motion with respect to the object v = velocity of moving object c = speed of light

Space-Time Curvature The concepts discussed above fall under the theory of “Special Relativity” because it only works when you’re moving uniformly in a straight line, which is where the Principle of Relativity applies. It doesn’t work in the cases where the object is accelerating.

Gravity Slows Down Time You already know that objects under acceleration which is NOT parallel to their motion follow a curvilinear path. Common examples include projectile motion and circular motion. Therefore, it is reasonable to expect that the path of light, which has dual nature of wave and particle, should also be curved in presence of acceleration. We also know that gravity induces acceleration on all objects. Therefore, gravity curves the path of light. But acceleration (or retardation) doesn’t just change the direction of movement of light, but also changes its velocity.

Imagine there’s a huge furry ball rolling past you at a very high speed and you try to grab it. If you can’t stop it completely, your effort will change the ball’s direction as well as slow it down. Therefore acceleration/gravity slows down light, i.e. it slows down light clocks, i.e. Gravity slows down time. This phenomenon has been experimentally verified by sending clocks round the Earth on jet planes which fly far above the earth; the deviations, although extremely small, have been detected accurately and they match the predictions made by general relativity. In the case of black holes; which are a common term in everyday language; the gravity is so strong that it doesn’t slow time – it stops it. It’s as if your effort in the aforementioned example stopped the ball. Thus, a paradox arises – if you watch someone approach a black hole in a spaceship, you’d see the ship get slower and slower. If the astronaut hung around the black hole for a while and then returned to you, you’d find less time has passed for them than for you. Of course, in its true sense, since time is relative and depends upon the velocity of the observer, it doesn’t make sense to compare the two times.

Gravity Curves Space Imagine a bicycle wheel spinning round and round at half the speed of light. Because the tyre is moving so fast, it must get shorter (due to Lorentz length contraction). A shorter tyre means a shorter wheel. But what about the spokes? They’re moving too, but only sideways, so they don’t get shorter, just thinner. How can the wheel get smaller yet stay the same size? The explanation is provided by the general theory of relativity. Einstein said that acceleration/gravity affects both space and time. The only way of shortening the tyre without shortening the spokes is by bending space so that there is a bit extra space inside the wheel, and the spokes have more room. Therefore acceleration/gravity bends space. Since gravity is in the first place caused by matter, therefore we can combine the above statements to: Matter bends space and time. Left to themselves, light-rays travel in straight lines. But objects of considerable mass, like stars, bend space-time. So now light cannot travel in a straight line, but has to travel along the curve of space time. This can imply startling results. Light from distant objects is bent by matter and so the observed position of the light source is different from its actual position.

Change in apparent position of stars due to space- time curvature

This can also be explained with the following example: Consider two diametrically opposite points on a torus. The shortest distance between them in 3 dimensions would be the straight line joining them. But for 2 dimensional beings on the surface of the torus, the shortest distance would be along a curve. A similar case arises when we observe our

surroundings in 3 dimensions; the motion might look complicated, but actually in a higher number of dimensions; it still moves in a straight line

Mass-Energy Equivalence Relativity showed that at the speed of light, clocks on board would stop and the object’s length would be zero. But could anyone actually go that fast? Let’s assume that you have a little machine that can measure the speed of objects. You flick a grain of sand at a wall at 90% speed of light. The sand makes a loud BANG when it hits the wall, and leaves a dent in it.

Now, say you flick it twice as hard. You’d expect it to go at 180% the speed of light. But your machine tells you that it only goes at 97.2% the speed of light. But it does make a louder BANG when it hits the

wall. The wall shakes about a bit and a few cracks appear. Now you try really hard and flick the sand grain twenty times harder than you did the first time. The machine will tell you that you’ve still only managed to get the grain up to 99.97% the speed of light. But when it hits the wall…

Relativity explains this by saying that the energy provided to the body increases its momentum but not its velocity, i.e. the mass of the body appears to increase and is known as relativistic mass. In fact

𝑚′ =

Where,

𝑚 𝑣2 1− 2 𝑐

m’ = relativistic mass m = mass when object is at rest with respect to the observer v = velocity of moving object c = speed of light

Albert Einstein, on a paper only three sides long, used relativity to explore what happens to a glowing object as it moves, and discovered a way of relating the energy of the glow to the mass of the object. His discovery applied to any energy (not just light) and any object (not just a moving one) and it led him to the most famous equation in the world

𝐸 = 𝑚𝑐 2 This implies that even miniature objects actually have a lot of locked-up energy (since the mass is multiplied by 9x1016)

Conclusion Einstein’s theory of Relativity has wide varying applications. It helps to accurately determine the position of moving aircrafts which is vital to the safety associated with air travel. Scientists who live in the International Space Station have been known to get a few seconds younger when they return to Earth after having lived there for about six months. This is because they travel at high speed around the Earth a number of times during their stay in the ISS. E=mc2 helps to determine the amount of energy stored in a substance. It is used to determine the amount of energy that will be released by radioactive substances during nuclear fission in the nuclear reactor. It is also used to design atom bombs. Relativity has also given insights about time travel and its effects. It tells us how we have to travel with speed of light to stop time and faster than it if we have to go back.

Relativity has changed our life; and the way we look at things; and has shown that sometimes, common sense just doesn’t work.