What are the prerequisites for understanding Lorentz transformations?

In summary, Lorentz transformations are linear transformations of the Minkowski coordinates that mix space and time. They satisfy the condition of being orthogonal and have a determinant of +1, preserving the Minkowski unit. They are not a group on their own, but when combined with the space rotation group SO(3), they form the Poincare group SO(1,3), which contains all special orthogonal transformations on Minkowski spacetime. These transformations are a direct consequence of the ability for two observers to compare their space-time coordinates and the constancy of the speed of light for both observers.
  • #1
preet0283
19
0
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
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
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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  • #4
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.
 

1. What are Lorentz transformations?

Lorentz transformations are a set of equations used in physics to describe the relationship between space and time in different reference frames, specifically in the theory of special relativity.

2. Why are Lorentz transformations important?

Lorentz transformations are important because they allow us to understand how space and time behave at high speeds, and they are a fundamental part of the theory of special relativity, which has been verified by numerous experiments.

3. How do Lorentz transformations differ from Galilean transformations?

Lorentz transformations differ from Galilean transformations in that they take into account the constant speed of light and the relativity of simultaneity, while Galilean transformations only consider relative velocities and absolute time.

4. How do Lorentz transformations affect measurements of time and distance?

Lorentz transformations affect measurements of time and distance by causing them to be relative to the observer's reference frame. This means that measurements of time and distance can appear different to different observers depending on their relative motion.

5. Can Lorentz transformations be applied to macroscopic objects?

Yes, Lorentz transformations can be applied to macroscopic objects. However, the effects of these transformations are only noticeable at speeds approaching the speed of light. At everyday speeds, the differences are too small to be measured.

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