How Does the Poincare Group Relate to Special Relativity and Reference Frames?

[CyberShot]

I've read that it is also a Lie group. But what does that have anything to do with special relativity or different reference frames?


The wiki definition is

"a 10-dimensional noncompact Lie group. The abelian group of translations is a normal subgroup while the Lorentz group is a subgroup, the stabilizer of a point. That is, the full Poincaré group is the affine group of the Lorentz group, i.e. the Poincaré group is a semidirect product of the translations and the Lorentz transformations:"

...blahblahblah

Can anyone please cut the bs out for me and tell me in layman terms the gut of what it is?

 

[Hurkyl]

First line:

In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.

Also, read the section on Poincaré symmetry.

 

[CyberShot]

First line:

If I told you to sum up GR, you'd say (mass curves space-time). If I told you to ask me what a derivative of a function was you'd say (rise over run); now why is it that no layman definition exists for a Poincare group?

I just want to know an overview.

 

[Ben Niehoff]

The Poincare is just what he said: the group of symmetries of flat spacetime. It's the group generated by translations, rotations, and boosts.

 

[Vinni]

The Poincare is just what he said: the group of symmetries of flat spacetime. It's the group generated by translations, rotations, and boosts.

So its like opengl where I'm pushing and poping the matrix while I apply rotations, translations and boosts?

 

[WannabeNewton]

There isn't an easy way to explain it like that. To put it loosely, isometries are special types of diffeomorphisms, with regards to the metric, in that certain vector fields will generate the diffeomorphisms that keep the metric unchanged. You could say the isometries are sort of like the inherent symmetries of the space - time in question such as time translation in the minkowski case.

 

[Ben Niehoff]

An isometry is a rigid motion of a manifold onto itself. For example, rotating a sphere on its axis is an isometry.

A flat plane has three independent isometries: translations in X and Y, and rotations.

Flat 3-space has 6 isometries: translations in X, Y, Z, and rotations in each of the XY, XZ, and YZ planes.

Flat spacetime has 10 isometries: translations in X, Y, Z, T, rotations in each of the XY, XZ, and YZ planes, and boosts along the X, Y, and Z directions.

All the isometries of a given space together form a group, which essentially means that every isometry has an inverse which is also an isometry, and the composition of two isometries is an isometry.

 

[meopemuk]

CyberShot,

You can get an idea about the Poincare group by considering the set of all inertial observers, i.e., those observers which move freely in space with constant velocities and without rotations. Suppose you have one such observer. Then you can ask, which transformations can be applied to this observer, so that the new (transformed) observer is also inertial? Apparently, there are only 10 classes of transformations with this property. They are (3) space translations, (3) space rotations, (3) boosts or velocity changes, and (1) time translation. These transformations map the set of inertial observers on itself, so they form a group with respect to the operation of composition. This is the Poincare group.

Eugene.

 

[GrayGhost]

CyberShot,

Consider the 2 systems, t,x,y,z and T,X,Y,Z (where T is Tau). By Einstein in his 1905 OEMB, he obtained his spacetime transformation equations ...

T = g(t-vx/c²)
X = g(x-vt)
Y = y
Z = z

Where gamma (g) …

g = 1/(1-v²/c²)^1/2

If the above transformations (called the Lorentz transformations) are represented by matricies, then we have ...

T = |...-g...-β*g...0...0 |...| t |
X = |.-β*g...g...0...0 |...| x |
Y = |...0...0...1...0 |...| y |
Z = |...0...0...0...1 |...| z |

where β = v/c
where g = 1/(1-v²/c²)^1/2

As can be seen, because of the two -β*g terms, time T depends on both t and x, and distance X depends on both t and x. That is, space and time are mutually linked at the hip, ie fused.

Now, my question to the community here ... Is "the above specific matrix of coefficients" what would be considered the Poincare group (or Lorentz group)? IOWs, the Poincare group is the set of variable coefficients of the Lorentz transformation equations. Wouldn't this be a reasonable way to convey it's meaning? That it defines what we call the "Lorentz symmetry"?

Similarly ...

The old Galilean transformations were ...

T = t
X = (x-vt)
Y = y
Z = z

If the above transformations (called the Galilean transformations) are represented by matricies, then we have ...

T = |...1...0...0...0 |...| t |
X = |...-v...1...0...0 |...| x |
Y = |...0...0...1...0 |...| y |
Z = |...0...0...0...1 |...| z |

And that this set of coefficients would be called the Galilean group?

GrayGhost

 

[Phrak]

Wikipedia can be useless; talking to those who already know the material. The Poincare group consists of the

1) Lorentz group
2) the group of spatial rotations
3) translations in space and time

The matrix representation is ugly, and not enlightening, but to include translations such as
x' = x + x₀, it is 5x5.