Lorentz transformations

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preet0283
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can ne 1 explain 2 me the basics of lorentz transformations...mathematically i know how things transform bt i want a more revealing explanation ...relate it 2 boosts and rotations also ...
thanx
 

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  • #2
selfAdjoint
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Lorentz transformations are linear transofrmations of the Minkowski coordinates that mix space and time. They are orthogonal transformations such that [tex]\Lambda \Lambda^T = \mathbf I[/tex]. And their determinants are +1, so they preserve the Minkowsi unit [tex]-c^2t^2 + x^2 + y^2 + z^2[/tex]. They do not form a group because the product of two of them can involve a spatial rotation; so you have to adjoin the space rotation group SO(3) to get the Poincare group SO(1,3). These are then all the special orthogonal transformations on Minkowski spacetime.
 
  • #3
pmb_phy
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selfAdjoint said:
Lorentz transformations are linear transofrmations of the Minkowski coordinates that mix space and time. They are orthogonal transformations such that [tex]\Lambda \Lambda^T = \mathbf I[/tex]. And their determinants are +1, so they preserve the Minkowsi unit [tex]-c^2t^2 + x^2 + y^2 + z^2[/tex]. They do not form a group because the product of two of them can involve a spatial rotation; so you have to adjoin the space rotation group SO(3) to get the Poincare group SO(1,3). These are then all the special orthogonal transformations on Minkowski spacetime.

Correct me if I'm wrote but Lorentz transformations are not orthogonal transformations since they do not satisfy the orthogonality condition that you stated above. An orthogonal transformation is defined as any transformation A which staisfies the relation AAT = I. Lorentz transformations satisfy don't satisfy that relation. They do, however, satisfy the relation LNLT = N where N = diag(-1, 1, 1, 1)

Pete
 
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robphy
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Technically, I guess one should prefix lots of terms by "pseudo-" (or "Minkowski-") when generalizing Euclidean concepts. However, after a while, we learn to generalize the concept to the non-euclidean case.

Technically, boosts and spatial rotations are examples of a Lorentz Transformation. It's the boosts that don't form a group. Finally, you have to adjoin the translations to the [Proper] Lorentz Group SO(3,1) to get the "inhomogeneous Lorentz Group", a.k.a. the Poincare group, ISO(3,1).
 
  • #5
preet0283 said:
can ne 1 explain 2 me the basics of lorentz transformations...mathematically i know how things transform bt i want a more revealing explanation ...relate it 2 boosts and rotations also ...
thanx

If you are looking for a non mathematical answer to your question, it's also possible to say that Lorentz transformations are a direct consequence of two pre-requizites: 1°) Two observers must have the possibility to compare their space-time coordinates via a linear transformation (and not via a bilinear one); 2°) speed of light (in vacuum) must appear to be the same for both observers if each of them is at the origin of what he calls an inertial frame. These two conditions are sufficient one to (for exemple) find the special formulation of the Lorentz transformations.
 

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