Special Relativity Derivation

In summary, unscientific is trying to derive eq 4.8 from the other two equations, but he doesn't understand what the RHS of 4.8 might look like. He suggests expanding 4.8 to understand how he can do the whole derivation starting from 2.27 and 2.28.
  • #1
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Taken from Steane's "Relativity made relatively easy" equation 4.8

2h3bigj.png


I have been trying to show (4.8) using these relations earlier on in the book:

2u5ymw4.png


Tried most means (rearranging, taking dot products) but can't seem to make it work. Is there an easy method I'm missing out?
 
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  • #2
Try expanding it out in the basis where x is the "u parallel" direction.
 
  • #3
DaleSpam said:
Try expanding it out in the basis where x is the "u parallel" direction.
Sorry, I don't understand what you mean by "it". Which term should I expand?
 
  • #5
DaleSpam said:
Expand both sides of 4.8.
I'm trying to derive eq 4.8 starting with the ones below, 2.27 and 2.28.
 
  • #6
Then expand all 3. The point is to get them all into the same form.
 
  • #7
DaleSpam said:
Then expand all 3. The point is to get them all into the same form.
I wouldn't know what RHS of eq 4.8 would look like initially. I am trying to derive its form starting from 2.27 and 2.28.
 
  • #8
Often, when you want to find a derivation from A to B, it helps to start by working from both A and B to find some equation in the "middle", so to speak. Then you can reverse the steps that took you from B to the "middle." Or, the working might give you enough insight that you can come up with a better way to write the complete derivation.
 
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  • #9
jtbell said:
Often, when you want to find a derivation from A to B, it helps to start by working from both A and B to find some equation in the "middle", so to speak. Then you can reverse the steps that took you from B to the "middle." Or, the working might give you enough insight that you can come up with a better way to write the complete derivation.

Point is, I wouldn't even know what RHS of 4.8 would look like.

OK here is the challenge, rephrased:

Without manipulating RHS of 4.8, use eq 2.27 and 2.28 to show that form.
 
  • #10
unscientific said:
Without manipulating RHS of 4.8,
Why would you add that restriction?

Just expand it. I don't know why you keep on saying that you don't know what the RHS of 4.8 is. u' and v are vectors, just write them as components and expand in the basis where x is the parallel direction.
 
  • #11
I believe unscientific doesn't want to manipulate the RHS of 4.8 because he considers that he isn't supposed to know what it is before deriving it from 2.27 and 2.28 in the first place.

@unscientific: even though you want to derive 4.8 directly from the other two equations, expanding 4.8 will help you understand how you can do that, because some steps may not be obvious. Then you can do the whole derivation starting from 2.27 and 2.28
 
  • #12
It may help to note that the parallellity (is that a word?) and perpendicularity are with reference to the relative velocity of the moving frame. In that case, u.v=u||×v + u×0.
 
  • #13
Start from:

[itex] \frac{1}{1 - (u \cdot v)/c^2 } = ...[/itex]
and you will reach the result. It's straightforward.
 
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  • #14
Thanks, got it!
 

1. What is special relativity and why is it important?

Special relativity is a theory proposed by Albert Einstein in 1905 that explains the relationship between space and time, and how they are affected by the motion of an object. It is important because it has revolutionized our understanding of the universe and has led to many technological advancements, such as GPS systems and nuclear energy.

2. How is special relativity derived?

Special relativity is derived from two main principles: the principle of relativity, which states that the laws of physics are the same for all observers in uniform motion, and the principle of the constancy of the speed of light, which states that the speed of light is the same for all observers regardless of their relative motion. From these principles, mathematical equations can be derived to explain the effects of time dilation and length contraction.

3. Can you explain time dilation and length contraction in special relativity?

Time dilation refers to the slowing down of time for an object in motion compared to an observer at rest. This is due to the fact that the speed of light is constant, so as an object approaches the speed of light, time for that object appears to slow down. Length contraction, on the other hand, refers to the shortening of an object's length in the direction of its motion. This is also a result of the constant speed of light, as an object's length appears to contract when it is moving at high speeds.

4. How does special relativity affect our everyday lives?

Special relativity has many practical applications in our everyday lives. For example, the theory is crucial for the functioning of GPS systems, which rely on precise measurements of time and space. Special relativity also helps us understand and predict the behavior of particles at high speeds, leading to advancements in nuclear energy and particle accelerators.

5. Are there any limitations to special relativity?

While special relativity is a widely accepted theory, there are some limitations to its applicability. It does not take into account the effects of gravity, which is addressed in Einstein's theory of general relativity. Additionally, special relativity only applies to objects in constant, straight-line motion, and cannot fully explain the behavior of objects in accelerated motion or at the quantum level.

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