If v = c then
c/c is 1 and 1-1 is 0
The Lorentz transform for length contraction is:
u = relative velocity between two reference frames.
Again, we have a situation where if relative velocity or difference in speed between two reference frames is equals c then l’ is equals zero.
THE TWIN PARADOX
Alison works for NASA. She is the creation of my imagination in my book so I can do whatever I want with her. She attended this business meeting as her twin sister Caroline and herself were chosen to travel in space.
I forgot to mention that it is a story that takes place far away in the future when mankind has developed to the point that they have a spaceship travelling at the speed of 3/5 of the speed of light. People also construct space stations at different distances from the Earth. Somehow they build the space station, which is exactly 3 light years away from Earth.
Both twins are thirty years old and were preparing for this journey. However, at the time of departure Caroline became unwell so it was only Alison who joined the crew. She therefore left Earth at the speed of 3/5 of the speed of light and travelled 5 years until her ship reached the space station. She returned back to Earth at the same speed of 3/5 of the light speed and it took her again 5 years to reach Earth.
Picture 6.10
Caroline continued her life on Earth for 10 years. From her point of observation in her reference frame at rest, the time in the spaceship where Alison is passes slower than for Caroline on Earth. We can imagine that for this purpose the Earth is at rest in relation to the spaceship. Caroline is therefore not moving through space but is moving through time. Even if you stand still, time is passing by. On the space-time diagram she is moving only along a vertical line representing time, as she has a zero movement in space, on the horizontal line. Alison is moving at 3/5 of the speed of light according to Caroline’s reference frame so she sees her sister, Alison, as aging more slowly than her. However, Alison sees Earth as moving away from her and perceives her ship as being at rest. From her reference frame, Caroline is the one who is aging more slowly. If Alison continues to travel until she dies then both sisters will perceive that it is the other one which is aging more slowly. As this will involve Alison travelling in one direction constantly until she dies, well passing the station at the distance of 3 light years, then the twins will never meet. There will not be the twin paradox, as they will never face each other.
Alison was, however, travelling in one direction for 5 years to the space station and was then instantly moving in the opposite direction for 5 years back to Earth.
The question is: which twin will be younger when they meet after 10 years?
Each of them thinks that the other one has aged more slowly. They cannot both be right.
The answer to this question is that Alison did not travel in one direction all her journey. After 5 years of travelling, her ship changed direction to return to Earth. The changing of direction of velocity, even if a constant speed, means that acceleration is happening. Alison is in this scenario really moving according to the reference frame of Caroline.
Why does acceleration determine that the one who accelerates will be in the reference frame where time slows down (time dilatation)?
The answer to that question is easy. Imagine that two spacecrafts move at the same speed of 3000 km per hour. Both of them will be in the same reference frame at the same speed and will not move in relation to each other. Now if one of them accelerates then it will really move in relation to the other and will be therefore the one where time dilatation will be taking place and the crew of this ship will age slowly according to the ship that does not accelerate but remains travelling at the speed of 3000 km/hour.
We can now work out how old Alison will be when she returns to Earth. Caroline will age ten years and will be forty years old. Both of them were thirty when Alison left Earth speeding at 3/5 of light speed towards the space station. Her age upon return can be calculated by using the Lorentz transform, which is:
t is Caroline time while t’ is Alison time.
v is her 3/5 of the light speed. So we can write fraction divided by fraction. In such mathematical calculations, it is the rule to multiply the outside number and put this product above the line. The product of multiplication of inner numbers comes under the line to make it only one fraction:
So:
Therefore:
So:
Or:
Therefore: 10=10 times 4/5 is 10=40/5, which is 10=8
While Caroline will have aged ten years and will be forty years old, Alison for the same time will age only eight years and will be thirty-eight years old.
MINKOWSKI SPACE TIME
Herman Minkowski was a German mathematician who came upon the idea of combining three space dimensions and one time dimension into four-dimensional space-time called ‘ Minkowski space’ or space-time which provided the mathematical foundations for Albert Einstein’s special theory of relativity.
In relation to so-called non-Euclidian geometry, Euclid was a mathematician who lived in Alexandria, Egypt. He was born in 300 BC. He came up with five famous postulates, which are also known as axioms, which contributed significantly to our understanding of geometry related to flat geometry. In this geometry, the shortest distance between two points is a straight line. Based on this geometry, the sum of the angles inside a triangle is 180 degrees. Also based on Euclidian geometry, which is referred to as flat geometry, the distance between two points can be found by applying Pythagoras’ theorem. If we draw lines through each point (A) and (B) so that these lines meet each other at the right angle then we can work out the value of the distance between A and B which will be the hypotenuse. Hypotenuse squared is equal to the sum of squared sides.
Picture 6.11 Δis a Greek symbol used to mark the difference in value as the difference between X1 and X2 or Y1 and Y2 in the above illustration.
On a circle, however, or on a square, the shortest line between A and B is not a straght line but a curved line. As it is not a straight line, it is rather called geodesic which defines the shortest part between A and B. Also, the sum of the inner angles of the triangle is not equal to 180 degrees but is more. If we try to draw a triangle starting from the North Pole and try to make a triangle of equal sides taken from the North Pole to the equator, then we can easily see that the angle between each side is 90 degrees, making the sum of the inner angles to be 270 degrees.
Picture 6.12
To go back to Minkowski space-time, as he combined three dimensions of space and one dimension of time he got an equation where hypotenuse squared is equal to time squared minus space squared.
S is the symbol for space-time interval or separation in time and space. Their value depends on the speed or velocity at which a particular object or event is in relation to the observer. However, when calculated using the above equation, then the space-time interval does not change and is the same regardless of another reference frame or place from where it is observed. It is obviously for the particular reference frame or velocity the particular object is travelling. Because it is the same, it presents invariance obtained by using this calculation. In the next illustration is an example of space-time separation at velocity of the speed of light. As we can see, we have a time of 3 years and distance light passed for 3 years (3 light years). As S is equal time t minus space x we have S equals zero.
We have therefore space-time interval or space –time separation zero at the speed of light. We can just remind ourselves of the example of the girl who is in the train speeding at the speed of light at 300 000 km/second and looking at herself in the mirror. The man on the ground was able to see light (photons from her face) going at the speed of 300 000 km/second as well as the girl on the train because the space between the mirror and the girl was zero as well as the time passed for photons leaving her face to reach the mirror. Thi
s is exactly what we get by applying the above equation to calculate the space-time interval or separation for an inertial frame of reference which travels with the speed of light. In other words, the space-time interval or separation is zero.
The trajectory of a movement at a certain speed in the space-time diagram is called the world line to differentiate it from using the term trajectory. The reason for this is that the world line marks the movement of an object in a space-time diagram not only in three-dimensional space where the term trajectory is used. The space-time separation for a particular world line is the result of time (t) minus space (x). We have a positive result for all world lines referreing to an object travelling at a speed less than the speed of light. That refers to any world line, which is between time coordinates ct or t and 45 degrees of light world line. Such space-time interval is called timelike as time separation is greater than space separation.
If an object takes light like a world line travelling at the speed of light, then space-time separation is zero. This object has to be massless, as we know that relativistic mass is the sum of rest mass and the speed. If an object has a mass then the mass will increase with the speed, and with it, energy needed to increase the speed of the object. If we want to push the speed of the object to the speed of light then its mass will increase infinitely and with it the energy needed to push this object to the speed of light. That is why an object has to be massless to be able to travel at the speed of light. In other words, an object that has mass at rest will never reach the speed of light as it requires infinite energy due to relativistic mass increasing towards infinity as its speed increases towards light speed.
If the space-time separation is a negative result then world lines are based between light world line of 45 degrees and the horizontal axis x presenting space separation. Such world lines are called spacelike as space separation is greater than time separation.
Picture 6.13
One important quality of constant and limited maximum speed in our universe, which is preserved for light speed, is that limitation of maximum speed presented by light speed secured the causality principle in the universe. This means that we can never encounter effect before its cause. Cause always has to precede the effect. Perhaps the simplest way to explain this is with the help of the illustration below:
Picture 6.14
A supernova has exploded at point A and has produced a gamma ray burst which travels at the speed of light, reaching the planet at point B. The result is the planet’s destruction.
The light from point A at the moment of the supernova explosion and the creation of the gamma burst reaches us at a particular time.
Light reflecting the destruction of the planet at point B will reach us at a particular time but will never be at the same time with the light from point A at the moment of explosion and the creation of the supernova and gamma rays. The reason is that light has a speed limit and it will take time to pass the distance from A (cause) to B to carry information which will result in the destruction of the planet (effect). The distance between A and O is less than the distance between A plus B plus O. We cannot have information about planet destruction before this information is delivered to point B from point A. All of this shows that cause is always seen first or chronologically it happens before effect. Therefore, the causality principle is secured by constant and limited maximum speed represented by the speed of light.
Simultaneity is another important event taking place in the universe, which can be perceived differently.
If we are referring to an event which is taking place at the same time and in the same place, then it will be seen as simultaneous regardless of the distance from which it is observed in space and time or in which reference frame is an observer. An example of this is a traffic accident when two cars collide, clapping with fingers or kissing. All these events take place at the same time and same place.
If we are talking of two traffic accidents happening at different places but at the same time then their simultaneity depends on the inertial reference frame an observer is in. Simultaneity is therefore relative. To understand this we should remind ourselves of the two main postulates of the special theory of relativity. In the first, it is stated that all laws of physics remain the same in every reference frame.
If I am travelling with the train I consider myself to be at rest. If I put myself in a time- space diagram I will move only along vertical axes representing time, as I believe that I am at rest and not moving in a horizontal line or through space. If two balls fall from a luggage compartment at the top of my carriage, one at the front and one at the back of the compartment at the same time, then I will perceive two events happening simultaneously. This is, however, in relation to my space axis which is at the angle with the time axis of my time in my referential frame. The proper time is the time shown by the clock, which I carry with me in my referential frame. This has invariant property upon which everybody agrees.
The second postulate is that the speed of light is always the same regardless of the reference frame of the observer (whether the observer is at rest or is at constant velocity).
To explain the above better, I believe that the next few illustrations will be helpful.
First, let’s imagine an observer is at rest. This is shown in the next diagram. There are also three events taking place, which are for this observer (being at rest) simultaneous.
Picture 6.15
In the above diagram concentrate for a moment only on the left-hand diagram and disregard the right-hand one to avoid confusion. The left-hand diagram is a diagram of an observer at rest who moves through his time along vertical time axes. At one point he will observe a simultaneous event taking place at A, B and C. It is according to his space axis, which is at 90 degrees with his time axis. We should not forget that light speed is taking a diagonal line, which is equally 45 degrees from time and space axes.
Forget for the moment the right-hand side of the diagram representing an observer in movement at constant velocity and concentrate on the next diagram. In the next diagram I am trying to draw a diagram of an observer who is moving at constant velocity CT’. We should not forget that this is a diagram of an observer who is moving at constant speed, combined with a diagram of the observer who is at rest. We have here two diagrams fused together. One, of an observer at rest CT (time axis) – X (space axis), and the other, of an observer who is moving at constant speed CT’ (time axis) X’ (space axis). He carries his watch with him, which shows him a proper time for his frame of reference where he considers himself at rest. He therefore moves only along his time axis CT’. However his time axis is moving with constant velocity in relation to the diagram of the observer who is at rest. That is why CT’ will be at an angle compared with CT. Now if we want to construct his space axis (X’), it should be at a right angle with his time axis (CT’) as shown in the right-hand diagram. This, however, is not possible because light will be going at C’. Although this would be at 45 degrees for his frame CT’ and X’ which is the speed of light, it will not be from the reference frame of the man at rest in the diagram.
Picture 6.16
In the reference frame of the observer at rest, C’ will be faster than the speed of light as it is less than 45 degrees with his X or space axes. The second postulate states that the speed of light is the same for all reference frames. We should notice that the speed of light forms the same angle with CT and X axis and that should be in all reference frames. We also need to have the speed of light at the same place in all reference frames. Therefore, the logic conclusion is that X’ should be at the same angle with the line of the speed of light like it is CT’ as in the next diagram.
Picture 6.17
Now the space axis of X’ is at the angle with the space axes of an observer in a referential frame at rest. If an observer in constant velocity at CT’ considers he is at rest taking his CT’ as his proper time axis then only events happe
ning in his space axis X’ will be simultaneous for him (the balls falling from the compartment in his carriage at the same time in the front and at the back of his carriage). However, events which take place simultaneously for an observer at rest (A B and C) in the first diagram on the left will not be simultaneous for an observer at reference frame CT’ as his X’ axis will be at an angle with the X axis. Now concentrate on the right-hand side of the illustration and diagram of an observer who is moving and in the reference frame CT’. He will observe first C, then B, and finally event A. To make it easier, I’ll repeat this illustration below. You can see that CT’ is at 35 degrees with the light speed world line and the same angle of 35 degrees is between the light speed world line and X’ axis.
Journey Through Time Page 14
