Exploring Lorentz Transformation Through Successive Boosts

  • Thread starter actionintegral
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In summary, ActionIntegral pointed me to the first thing I needed to understand the Lorentz transformation - which is to find the generator of the group. Hi, ActionIntegral, you were correct - the generator is the element that behaves like a function that increases by a certain amount each time it's applied. Thanks to all of you for participating!
  • #1
actionintegral
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5
Hi Friends,

From time to time I have seen transformations replaced by a succession of infinitesimal transformations. The end result ends up being an exponent.

My knowledge of this is vague and I would like to look into it more seriously. Particularly I am interested in describing the lorentz transformation as a sequence of infinitesimal boosts.

Can someone point me to the first thing I need read to understand this?

Thanks
 
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  • #2
Hi, ActionIntegral,

I'll be more than happy to take a stab at it! Just expand the lorentz transform in a taylor series in v about v=0 keeping only the first order.

You can raise this operator to the nth power. Somehow this becomes an exponent but I haven't figured that part out yet.
 
  • #3
actionintegral said:
Hi, ActionIntegral,

Somehow this becomes an integral but I haven't figured that part out yet.

That's ok - I wouldn't want you to do all the work for me! :smile:
 
  • #4
actionintegral said:
Hi Friends,

From time to time I have seen transformations replaced by a succession of infinitesimal transformations. The end result ends up being an exponent.

My knowledge of this is vague and I would like to look into it more seriously. Particularly I am interested in describing the lorentz transformation as a sequence of infinitesimal boosts.

Can someone point me to the first thing I need read to understand this?

Thanks

What I think you're trying to do is find the "generator" of the Lorentz group.

http://en.wikipedia.org/wiki/Generating_set_of_a_group

In abstract algebra, a generating set of a group G is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses.

But you want to do this for a continuous group. The Lorentz transformation is a "group" in the abstract algebra sense with the group operation being the successive application of transforms, because the result is associative f x (g x h) = (f x g) x h, and has an inverse. However, the Lorentz group is an infinite group. This is called a "Lie group".

So what you need to do is to read up on Lie groups (specifically the generators of Lie groups). Or see if you can find a mathemetician.

Hope this helps.
 
  • #5
Let [tex]\Theta=\left[ \begin {array}{cc} 0&1\\\noalign{\medskip}1&0\end {array} \right]\theta[/tex].
Formally, write [tex]\exp(\Theta)=I+\Theta+\Theta^2/2!+\Theta^3/3!+\ldots[/tex]. Do you recognize [tex]\exp(\Theta)[/tex]?
 
  • #6
Both good answers - i'll read up on it. thanks to all three of you!
 

FAQ: Exploring Lorentz Transformation Through Successive Boosts

1. What is the Lorentz Transformation?

The Lorentz Transformation is a mathematical formula used to describe the relationship between space and time in special relativity. It allows for the conversion of measurements between different frames of reference moving at constant velocities.

2. How do successive boosts affect the Lorentz Transformation?

Successive boosts refer to the application of multiple Lorentz Transformations, each representing a change in velocity between frames of reference. These boosts can have a compounding effect on the transformation, resulting in a more complex mathematical formula.

3. What is the significance of exploring Lorentz Transformation through successive boosts?

By exploring Lorentz Transformation through successive boosts, we can gain a deeper understanding of how changes in velocity affect measurements of space and time. This is particularly relevant in the field of special relativity and has real-world applications in areas such as GPS technology.

4. What are some examples of successive boosts in real life?

One example of successive boosts is the Doppler effect, which occurs when an object emitting waves, such as sound or light, is moving relative to an observer. Another example is the redshift of light from distant galaxies, which is caused by the expansion of the universe and successive boosts as the light travels towards us.

5. Are there any limitations to exploring Lorentz Transformation through successive boosts?

While exploring Lorentz Transformation through successive boosts can provide valuable insights, it is important to note that this approach is based on simplifying assumptions and may not accurately represent all real-world scenarios. Additionally, the calculations involved can become increasingly complex with each successive boost, making it difficult to accurately analyze in some cases.

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