Triangles And Time Warps
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“Triangles and Time Warps”
By, David.R.Gilson, MPhys, AMInstP http://www.davidgilson.co.uk Copycenter 2006 (http://en.wikipedia.org/wiki/Copycenter)
Name:…………………………………………………………
“Triangles and Time Warps”
Einstein’s thought experiment The inspiration that lead Albert Einstein to create his “theory of (special) relativity” is a very simple idea that makes us all realise that our everyday perception of relative motion is incorrect. When we are talking about relativity we often talk about “Frames of reference”. These are simply some words to discriminate between observers (i.e. people) that are moving relative to each other. In a previous session we discussed how two observers, one stationary (relative to us!) and one moving (relative to us) would measure a completely different speed for the same object because they themselves have different relative speeds. For instance, consider three people, A, B and C and they are all equipped with speed measuring devices. Person A is stood still on the roadside, and persons B and C are driving cars. Person B is travelling at 30mph and Person C is travelling at 60mph.
Person B 30 mph
Person C 60 mph
Person A
Person A measure the same speeds as we do since we are stationary to person A. However, Person B would only measure Person C moving forwards at 30 mph and would see Person A moving backwards at 30mph too. Person C would measure Person B moving backwards at 30mph and Person A moving backwards at 60mph! This is our everyday notion of relative motion. About a century ago, a physicist named James Maxwell wrote a famous set of equations about light. He deduced that the speed of light was a fixed number, unlike a car where you can vary the speed by pressing the accelerator or the brake pedals. In any given material there is nothing that can change the speed of light, it is a law of nature. Going back to our picture with the cars, whether we are standing still, driving at 30mph or 60mph, it is reasonable to assume that the laws of nature are the same everyone. Just because you are moving faster or slower than someone else does not mean that the laws of nature will change for you. For Einstein this raised a worrying question: what if Person’s A, B and C were all to try measuring the speed of light? Because they all have different relative speeds to each other; Person A and B never agree on Person C’s speed, and so on. However, the
Page 2 of 9
“Triangles and Time Warps”
speed of light is a law of nature for which everyone must agree, regardless of how fast you are moving! ? Person B 30 mph LIGHT
?
Person C
? 60 mph
Person A
There has to be something wrong in our everyday experience of space and time and relative motion because we cannot explain why person A and B do not agree on the speed of person C but they do agree on the speed of light! So our question is this: “how can all frames of reference disagree on relative speeds of other frames of reference, yet they all agree on the speed of light?” We are now going to take this physics based idea and apply mathematics step by step to investigate and solve our problem! First of all though, we need to look in our mathematical box of tricks and make sure we have the right tools for the job!
Page 3 of 9
“Triangles and Time Warps”
Mathematical tools Here is a list of mathematical techniques that will be used as we tackle our problem, you may need to refer back to this page to help you perform each step of our investigation!
Factorisation You can think of factorisation as the opposite to multiplying out brackets. Here are some examples of factorisation (remember that in algebra “AB” means “A multiplied by B”).
AB + AC = A( B + C ) and AB − AC = A( B − C )
5 ∗ 4 + 5 ∗ 3 = 5 ∗ ( 4 + 3) = 35 and 5 ∗ 4 − 5 ∗ 3 = 5 ∗ ( 4 − 3) = 5 In these examples going from left to right is factorisation and going from right to left is multiplying out brackets. When factorising you simply divide all terms by a common factor that you place outside the brackets. With that in mind here is another example of how you can factorise: ( A − B ) = A(1 − B A) (Jargon Buster: “Common Factor” means a number/variable that all terms are multiplied by, i.e. it is a factor that is common to all, in the numerical example above 5 is the common factor).
Squares and square roots You will have come across these before, but you might not have used them together. Taking squares and square roots are opposite operations, just like adding and subtracting are. If you take a number, square it, then take the square root of that number you return to your original number. Here are some examples.
B = A and also,
A 2 = AA and if B = A 2 then In other words,
AA =
A2 =
( A)
2
( A)
2
= A
= A
Square roots of products (Jargon Buster: a “product” just means, “some things multiplied together”!!!) If you have the square root of a product (see the jargon buster above), you can split the square root into a product of square roots. Here is a numerical example to prove this to you…
100 =
4 ∗ 25 =
4∗
25 = 2 ∗ 5 = 10 and in general we can write,
AB =
A B , A and B can be anything, say if we’d factorised something and taken square roots we could write C ( A − B ) = C ( A − B ) . The same applies when taking squares, so we can simply write ( AB ) = A 2 B 2 . 2
Simplifying Equations Factorisation involves dividing one side of an equation through by a common factor and placing that factor outside of brackets. If you have the same factor on both sides of an equation you can divide both sides by that factor to remove that factor from the equation. This is called simplifying. Here is an example:
LA LC A C = = will simplify to by dividing both sides by L. B D B D
Page 4 of 9
“Triangles and Time Warps”
The Investigation For our investigation we have an imaginary piece of equipment that we use to measure the speed of light called a “light clock”. It is simply a dark box with an emitter at the bottom and a detector at the top, which are both linked to a timer. We know the size of the box, so if we fire the emitter some light is sent from the bottom of the box to the top of the box and the timer is started. When the light hits the detector at the top the timer is stopped. Hence we have a distance and a time from which we can calculate the speed of light, given that speed=distance/time. Before we go on, it should be pointed out that the speed of light is universally given the symbol “c”! We also have two imaginary people, A and B. Both A and B have a light clock each. For our investigation we imagine that A stands still with us and B is moving past us in a fast car. We will call the time to get from bottom to top in A’s clock tA and we will call the time taken to get from bottom to top in B’s clock tB. Let’s look at A’s light clock, c
tA =00:00
L
Hence, after a time tA the light has travelled at a speed c and covered a distance L = ct A . Using what you know complete the two following equations using the quantities in the diagram above:
[ ]
c=
t = A [ ]
[ ] [ ]
That was fairly straight forward, now lets look at B, this is more complicated because of B’s sideway motion relative to us and A. B is moving sideways with a speed of v. Our first aim is to compute what tB is, and then use that to calculate the speed of light as you did above. When we watch the beam of light go from bottom to top in B’s light clock we see the light take a diagonal path because of the sideways motion.
c
v
L
Time=t B
Time=0
Page 5 of 9
“Triangles and Time Warps”
In the time it takes the light in B’s clock to get from bottom to top (being tB) B has moved a sideways distance of vtB and also in that time the beam of light travelled a distance ctB. Since the direction of B’s motion is at right angles to the height of the box we can now form a right-angled triangle.
ctB
L
vtB We can apply Pythagoras’s theorem here, fill in the blanks below:
[ ]2[ ]2 = [ ]2[ ]2 + [ ]2 Keep in mind that we are trying to compute tB, so the next logical step is to get all the terms containing tB on the same side, hence subtract v 2 t B2 from both sides and fill in the blanks below:
[ ] 2[ ]2 − [ ]2[ ]2 = [ ]2 t B2 is now a common factor on the left hand side so you can factorise it out, complete the following:
[ ] 2 ([ ] 2 − [ ] 2 ) = [ ] 2
We nearly have tB by itself, but it is still squared so now would be a good time to take the square root of both sides (look back at the part about square roots in our tool box on page 4 if you need to).
[ ] [ ]2 − [ ]2 = [ ] Now all we have to do to compute tB is to divide both sides by the square root term that appears on the left hand side above. You’ve guessed it, try filling in the blanks below!
[ ]=
[ ] [ ]2 − [ ]2
We have now found tB, using this equation we would like to rearrange it to give an expression for the speed of light, c, so that we can compare with A’s equation for c. However, in its present form the equation above does not give us easy access to c, so we need to use some more of our mathematical tricks to get at it! The speed of light appears in one place, within the square root. However it is possible to use factorisation to extract a factor of c2 then we can split the square root (see examples in the tool box!). With that in mind; fill in the blanks in the next equation.
Page 6 of 9
“Triangles and Time Warps”
L
tB =
[ ] 2 [ ] 2
c 1 − 2
We can now split the square root in to a product of two square roots (refer back to the toolbox for this). When we do this one of our square roots is of c squared, which if you look at the toolbox you will see just gives us c. Writing it our properly gives us; c 2 = c . So, we have finally extracted c by itself! If you follow this idea, try filling in the blanks below:
tB =
L
[ ]
[ ]2 [ ]2
1−
The square root term is a very important mathematical object in relativity, and it appears in almost every relativistic calculation that someone may perform, so much so we give it a special symbol, “γ”, which is a Greek letter called, “Gamma”. We define Gamma as:
1
γ =
1−
v2 c2
Therefore, we write tB as:
tB =
Lγ c
So we have finally computed tB in such a way that we can rearrange to give an expression for c. Remember the only reason we are doing this is because we want to compare equations of c for Person A (stationary) and Person B (moving). So, for both A and B we have the following equations:
c=
L tA
&
c=
Lγ tB
Now think about our original problem, the speed of light is the same for everybody! That means that the speed of light is the same for A and B, i.e. both c’s are the same thing! So we can now write a new equation:
L Lγ = tA t B Both sides of this have a factor of L, so we can simplify it now (see the toolbox):
1 γ = t A tB
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“Triangles and Time Warps”
We can multiply both sides by tA and then by tB to get rid of the fractions too:
tB = γ ⋅ t A This in itself is quite surprising, it shows us that time is different for A and B. We would expect everyone’s clocks to be the same from our everyday experience, but this is not so according to our final equation here. If you look back at the definition of Gamma, the only variable is v, the relative speed between A and B. They say a picture tells a thousand words, plotting a graph gives us a picture of an equation and is an invaluable tool. Below is a graph of Gamma versus relative speed, we can see how Gamma varies as v increases.
Gamma
Chart of the Gam m a factor ve rs us re lative s pe e d be tw e e n tw o fram e s of re fe re nce . (note that the s pe e d of light is approxim ate ly 300 m illion m e te rs pe r s e cond!)
18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
0
100
200
300
Relative speed (million-meters per second)
This shows us that gamma slowly increases as the speed between frames of reference increases. You should note the huge speeds required to cause a significant increase in Gamma. If we recall the equation for A and B’s time we can see that from A’s point of view time in B’s frame will appear to increase. In other words, from A’s frame of reference, time in B’s frame of reference will appear to slow down. This is because Gamma is greater than or equal to one, which means multiplying tA by Gamma guarantees that tB will always be larger than tA. Before we start to disbelieve this incredible result, we should study our chart. Notice how large a relative speed must be before a noticeable increase in Gamma is reached! Notice also at how the graph seems to be almost vertical and never quite passing through the speed of light threshold? This is a hint at a deeper result of relativity, that the speed of light is a kind of universal speed limit. To date, we know of nothing that
Page 8 of 9
“Triangles and Time Warps”
can travel faster than light, and indeed there is only light that can reach the exact speed of light. This is a very weird result, but it is real and has been tested in numerous experiments. You should also realise that from B’s frame of reference time is moving slowly for us because B observes A (and us) moving away at v as well. This effect of “moving clocks running slow” is what Einstein called “Time Dilation”. There are other relativistic effects that go along with Time Dilation. “Length Contraction” for instance is an effect where the more your speed approaches the speed of light you will observe length of objects outside of your frame of reference begin to shrink, or contract. There is much more to this rich subject, but I leave that for you to find out about those as you continue your studies.
Conclusion This is a fascinating exercise in itself, but discovering Time Dilation was not the main goal. What you have done in this exercise is taken on board reasoned problems using physics based concepts, and have then applied pure mathematics to model the ideas and have come up with a experimentally testable result. By simply using our mathematics we have discovered something far beyond our own experience of everyday life and learnt something new about the universe! Hopefully now you will see that mathematics is not as divorced from the real world as it can seem sometimes and that it can be a tremendously powerful tool!
Page 9 of 9
By, David.R.Gilson, MPhys, AMInstP http://www.davidgilson.co.uk Copycenter 2006 (http://en.wikipedia.org/wiki/Copycenter)
Name:…………………………………………………………
“Triangles and Time Warps”
Einstein’s thought experiment The inspiration that lead Albert Einstein to create his “theory of (special) relativity” is a very simple idea that makes us all realise that our everyday perception of relative motion is incorrect. When we are talking about relativity we often talk about “Frames of reference”. These are simply some words to discriminate between observers (i.e. people) that are moving relative to each other. In a previous session we discussed how two observers, one stationary (relative to us!) and one moving (relative to us) would measure a completely different speed for the same object because they themselves have different relative speeds. For instance, consider three people, A, B and C and they are all equipped with speed measuring devices. Person A is stood still on the roadside, and persons B and C are driving cars. Person B is travelling at 30mph and Person C is travelling at 60mph.
Person B 30 mph
Person C 60 mph
Person A
Person A measure the same speeds as we do since we are stationary to person A. However, Person B would only measure Person C moving forwards at 30 mph and would see Person A moving backwards at 30mph too. Person C would measure Person B moving backwards at 30mph and Person A moving backwards at 60mph! This is our everyday notion of relative motion. About a century ago, a physicist named James Maxwell wrote a famous set of equations about light. He deduced that the speed of light was a fixed number, unlike a car where you can vary the speed by pressing the accelerator or the brake pedals. In any given material there is nothing that can change the speed of light, it is a law of nature. Going back to our picture with the cars, whether we are standing still, driving at 30mph or 60mph, it is reasonable to assume that the laws of nature are the same everyone. Just because you are moving faster or slower than someone else does not mean that the laws of nature will change for you. For Einstein this raised a worrying question: what if Person’s A, B and C were all to try measuring the speed of light? Because they all have different relative speeds to each other; Person A and B never agree on Person C’s speed, and so on. However, the
Page 2 of 9
“Triangles and Time Warps”
speed of light is a law of nature for which everyone must agree, regardless of how fast you are moving! ? Person B 30 mph LIGHT
?
Person C
? 60 mph
Person A
There has to be something wrong in our everyday experience of space and time and relative motion because we cannot explain why person A and B do not agree on the speed of person C but they do agree on the speed of light! So our question is this: “how can all frames of reference disagree on relative speeds of other frames of reference, yet they all agree on the speed of light?” We are now going to take this physics based idea and apply mathematics step by step to investigate and solve our problem! First of all though, we need to look in our mathematical box of tricks and make sure we have the right tools for the job!
Page 3 of 9
“Triangles and Time Warps”
Mathematical tools Here is a list of mathematical techniques that will be used as we tackle our problem, you may need to refer back to this page to help you perform each step of our investigation!
Factorisation You can think of factorisation as the opposite to multiplying out brackets. Here are some examples of factorisation (remember that in algebra “AB” means “A multiplied by B”).
AB + AC = A( B + C ) and AB − AC = A( B − C )
5 ∗ 4 + 5 ∗ 3 = 5 ∗ ( 4 + 3) = 35 and 5 ∗ 4 − 5 ∗ 3 = 5 ∗ ( 4 − 3) = 5 In these examples going from left to right is factorisation and going from right to left is multiplying out brackets. When factorising you simply divide all terms by a common factor that you place outside the brackets. With that in mind here is another example of how you can factorise: ( A − B ) = A(1 − B A) (Jargon Buster: “Common Factor” means a number/variable that all terms are multiplied by, i.e. it is a factor that is common to all, in the numerical example above 5 is the common factor).
Squares and square roots You will have come across these before, but you might not have used them together. Taking squares and square roots are opposite operations, just like adding and subtracting are. If you take a number, square it, then take the square root of that number you return to your original number. Here are some examples.
B = A and also,
A 2 = AA and if B = A 2 then In other words,
AA =
A2 =
( A)
2
( A)
2
= A
= A
Square roots of products (Jargon Buster: a “product” just means, “some things multiplied together”!!!) If you have the square root of a product (see the jargon buster above), you can split the square root into a product of square roots. Here is a numerical example to prove this to you…
100 =
4 ∗ 25 =
4∗
25 = 2 ∗ 5 = 10 and in general we can write,
AB =
A B , A and B can be anything, say if we’d factorised something and taken square roots we could write C ( A − B ) = C ( A − B ) . The same applies when taking squares, so we can simply write ( AB ) = A 2 B 2 . 2
Simplifying Equations Factorisation involves dividing one side of an equation through by a common factor and placing that factor outside of brackets. If you have the same factor on both sides of an equation you can divide both sides by that factor to remove that factor from the equation. This is called simplifying. Here is an example:
LA LC A C = = will simplify to by dividing both sides by L. B D B D
Page 4 of 9
“Triangles and Time Warps”
The Investigation For our investigation we have an imaginary piece of equipment that we use to measure the speed of light called a “light clock”. It is simply a dark box with an emitter at the bottom and a detector at the top, which are both linked to a timer. We know the size of the box, so if we fire the emitter some light is sent from the bottom of the box to the top of the box and the timer is started. When the light hits the detector at the top the timer is stopped. Hence we have a distance and a time from which we can calculate the speed of light, given that speed=distance/time. Before we go on, it should be pointed out that the speed of light is universally given the symbol “c”! We also have two imaginary people, A and B. Both A and B have a light clock each. For our investigation we imagine that A stands still with us and B is moving past us in a fast car. We will call the time to get from bottom to top in A’s clock tA and we will call the time taken to get from bottom to top in B’s clock tB. Let’s look at A’s light clock, c
tA =00:00
L
Hence, after a time tA the light has travelled at a speed c and covered a distance L = ct A . Using what you know complete the two following equations using the quantities in the diagram above:
[ ]
c=
t = A [ ]
[ ] [ ]
That was fairly straight forward, now lets look at B, this is more complicated because of B’s sideway motion relative to us and A. B is moving sideways with a speed of v. Our first aim is to compute what tB is, and then use that to calculate the speed of light as you did above. When we watch the beam of light go from bottom to top in B’s light clock we see the light take a diagonal path because of the sideways motion.
c
v
L
Time=t B
Time=0
Page 5 of 9
“Triangles and Time Warps”
In the time it takes the light in B’s clock to get from bottom to top (being tB) B has moved a sideways distance of vtB and also in that time the beam of light travelled a distance ctB. Since the direction of B’s motion is at right angles to the height of the box we can now form a right-angled triangle.
ctB
L
vtB We can apply Pythagoras’s theorem here, fill in the blanks below:
[ ]2[ ]2 = [ ]2[ ]2 + [ ]2 Keep in mind that we are trying to compute tB, so the next logical step is to get all the terms containing tB on the same side, hence subtract v 2 t B2 from both sides and fill in the blanks below:
[ ] 2[ ]2 − [ ]2[ ]2 = [ ]2 t B2 is now a common factor on the left hand side so you can factorise it out, complete the following:
[ ] 2 ([ ] 2 − [ ] 2 ) = [ ] 2
We nearly have tB by itself, but it is still squared so now would be a good time to take the square root of both sides (look back at the part about square roots in our tool box on page 4 if you need to).
[ ] [ ]2 − [ ]2 = [ ] Now all we have to do to compute tB is to divide both sides by the square root term that appears on the left hand side above. You’ve guessed it, try filling in the blanks below!
[ ]=
[ ] [ ]2 − [ ]2
We have now found tB, using this equation we would like to rearrange it to give an expression for the speed of light, c, so that we can compare with A’s equation for c. However, in its present form the equation above does not give us easy access to c, so we need to use some more of our mathematical tricks to get at it! The speed of light appears in one place, within the square root. However it is possible to use factorisation to extract a factor of c2 then we can split the square root (see examples in the tool box!). With that in mind; fill in the blanks in the next equation.
Page 6 of 9
“Triangles and Time Warps”
L
tB =
[ ] 2 [ ] 2
c 1 − 2
We can now split the square root in to a product of two square roots (refer back to the toolbox for this). When we do this one of our square roots is of c squared, which if you look at the toolbox you will see just gives us c. Writing it our properly gives us; c 2 = c . So, we have finally extracted c by itself! If you follow this idea, try filling in the blanks below:
tB =
L
[ ]
[ ]2 [ ]2
1−
The square root term is a very important mathematical object in relativity, and it appears in almost every relativistic calculation that someone may perform, so much so we give it a special symbol, “γ”, which is a Greek letter called, “Gamma”. We define Gamma as:
1
γ =
1−
v2 c2
Therefore, we write tB as:
tB =
Lγ c
So we have finally computed tB in such a way that we can rearrange to give an expression for c. Remember the only reason we are doing this is because we want to compare equations of c for Person A (stationary) and Person B (moving). So, for both A and B we have the following equations:
c=
L tA
&
c=
Lγ tB
Now think about our original problem, the speed of light is the same for everybody! That means that the speed of light is the same for A and B, i.e. both c’s are the same thing! So we can now write a new equation:
L Lγ = tA t B Both sides of this have a factor of L, so we can simplify it now (see the toolbox):
1 γ = t A tB
Page 7 of 9
“Triangles and Time Warps”
We can multiply both sides by tA and then by tB to get rid of the fractions too:
tB = γ ⋅ t A This in itself is quite surprising, it shows us that time is different for A and B. We would expect everyone’s clocks to be the same from our everyday experience, but this is not so according to our final equation here. If you look back at the definition of Gamma, the only variable is v, the relative speed between A and B. They say a picture tells a thousand words, plotting a graph gives us a picture of an equation and is an invaluable tool. Below is a graph of Gamma versus relative speed, we can see how Gamma varies as v increases.
Gamma
Chart of the Gam m a factor ve rs us re lative s pe e d be tw e e n tw o fram e s of re fe re nce . (note that the s pe e d of light is approxim ate ly 300 m illion m e te rs pe r s e cond!)
18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
0
100
200
300
Relative speed (million-meters per second)
This shows us that gamma slowly increases as the speed between frames of reference increases. You should note the huge speeds required to cause a significant increase in Gamma. If we recall the equation for A and B’s time we can see that from A’s point of view time in B’s frame will appear to increase. In other words, from A’s frame of reference, time in B’s frame of reference will appear to slow down. This is because Gamma is greater than or equal to one, which means multiplying tA by Gamma guarantees that tB will always be larger than tA. Before we start to disbelieve this incredible result, we should study our chart. Notice how large a relative speed must be before a noticeable increase in Gamma is reached! Notice also at how the graph seems to be almost vertical and never quite passing through the speed of light threshold? This is a hint at a deeper result of relativity, that the speed of light is a kind of universal speed limit. To date, we know of nothing that
Page 8 of 9
“Triangles and Time Warps”
can travel faster than light, and indeed there is only light that can reach the exact speed of light. This is a very weird result, but it is real and has been tested in numerous experiments. You should also realise that from B’s frame of reference time is moving slowly for us because B observes A (and us) moving away at v as well. This effect of “moving clocks running slow” is what Einstein called “Time Dilation”. There are other relativistic effects that go along with Time Dilation. “Length Contraction” for instance is an effect where the more your speed approaches the speed of light you will observe length of objects outside of your frame of reference begin to shrink, or contract. There is much more to this rich subject, but I leave that for you to find out about those as you continue your studies.
Conclusion This is a fascinating exercise in itself, but discovering Time Dilation was not the main goal. What you have done in this exercise is taken on board reasoned problems using physics based concepts, and have then applied pure mathematics to model the ideas and have come up with a experimentally testable result. By simply using our mathematics we have discovered something far beyond our own experience of everyday life and learnt something new about the universe! Hopefully now you will see that mathematics is not as divorced from the real world as it can seem sometimes and that it can be a tremendously powerful tool!
Page 9 of 9
