Show that rotations and boosts lead to a combined boost

In summary, the conversation discusses the proof of applying an infinitesimal rotation followed by a boost and its inverse resulting in a single boost. The conversation involves finding the correct mathematical equation and simplifying it to prove this statement. Through the process of expanding the exponential of an infinitesimal rotation and finding a new G' term, the equation is eventually simplified and proven to be true.
  • #1
ma18
93
1

Homework Statement


Prove that applying an infinitesimal rotation of angle k<<1 around the axis x1, then a boost of speed -k along the axis x2 then the inverses of these is equal to a single boost of speed k^2 along the axis x3

The Attempt at a Solution



Putting this into mathematical terms I get

e-i*k*Jx1*ei*k*Gx2*ei*k*Jx1*e-i*k*Jx2 = e-i*k2*Gx3

However I don't really know where to go from here, the example we did in class only involved the rotations and we just used the matrix representations around the axis, but I don't know how exactly do it with the boosts.

Any help would be much appreciated.
 
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  • #2
You can expand the exponential of an infinitesimal rotation and then simplify.
 
  • #3
mfb said:
You can expand the exponential of an infinitesimal rotation and then simplify.
Doing this I get to the equation:

upload_2015-10-27_0-24-32.png


Which after expanding leads me to

upload_2015-10-27_0-27-30.png


Taking only the terms up to e^2 reduces this to

upload_2015-10-27_0-29-57.png


Which is not equal to the right hand side with only a single boost which expands to

upload_2015-10-27_0-31-21.png


What am I missing?
 

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  • #4
Where does e come from and where did k go?
Your boost is along a different directions, the G on the different sides are not the same.
Also, I think there is some error in the expansion of the right hand side.
 
  • #5
mfb said:
Where does e come from and where did k go?
Your boost is along a different directions, the G on the different sides are not the same.
Also, I think there is some error in the expansion of the right hand side.
Ah sorry, I used e in the equation instead of k, they are the same. I see, as the G is different what would be my next step?
 
  • #6
Find a new G' such that the equation is true up to order k2.
If you fix the expansion of the RHS, you are just one line away from this.
 
  • #7
ma18 said:
Ah sorry, I used e in the equation instead of k, they are the same.
mfb said:
Find a new G' such that the equation is true up to order k2.
If you fix the expansion of the RHS, you are just one line away from this.

Ah I see, the RHS is supposed to be (where G_x represents the new boost around x_3)

upload_2015-10-27_18-22-1.png


and then taking out the e^4 term this reduces to

upload_2015-10-27_18-22-42.png


and solving for G_x we get

upload_2015-10-27_18-23-14.png


Correct?
 

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  • #9
mfb said:
Should be right.
Great, thank you so much for your help
 

1. How do rotations and boosts combine to create a combined boost?

Rotations and boosts are both ways of transforming an object's position and orientation in space. When they are applied together, they create a combined boost, which is a combination of both types of transformations. This allows for more complex and dynamic movements in space.

2. What is the difference between a rotation and a boost?

A rotation is a transformation that changes an object's orientation or direction of movement without changing its position. A boost, on the other hand, changes both the orientation and position of an object. In other words, a boost is a combination of a rotation and a translation.

3. How do rotations and boosts affect the speed of an object?

Rotations do not affect an object's speed, as they only change its direction of movement. However, a boost changes an object's position and can therefore affect its speed. Depending on the direction and magnitude of the boost, it can increase or decrease an object's speed.

4. Can a combined boost be broken down into separate rotations and boosts?

Yes, a combined boost can be broken down into separate rotations and boosts. This is because a combined boost is essentially a combination of these two types of transformations, and can be broken down into its individual components.

5. How are rotations and boosts represented mathematically?

Rotations and boosts can be represented using matrices in mathematics. These matrices contain elements that describe the angle, direction, and magnitude of the transformation. By multiplying these matrices, we can combine rotations and boosts to create a combined boost.

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