Lorentz transformation invariant shape

[Edgard Neuman]

Hi,

 

I imagined a hyper sphere in spacetime (an spherical event) that would be invariant by Lorentz transformation.

Given a length R in a (x,y,z,t) space (around any event point) I considere the hypersphere given by :

 

 

x²+y²+z²+(t*c)² = R

 

Isn't it invariant by Lorentz transformation ?

Edited April 23, 2014 by Edgard Neuman

[xyzt]

Hi,

 

I imagined a hyper sphere in spacetime (an spherical event) that would be invariant by Lorentz transformation.

 

Given a length R in a (x,y,z,t) space (around any event point) I considere the hypersphere given by :

 

 

x²+y²+z²+(t*c)² = R

 

Isn't it invariant by Lorentz transformation ?

No, it isn't. You can prove that to yourself by transforming (x,y,z,t) into (x',y',z',t'). It is a simple algebra exercise.

[Edgard Neuman]

ok my mistake,

 

thanks !

Edited April 23, 2014 by Edgard Neuman

[ajb]

In general we don't have true rigid motion in special relativity. The closest thing to rigid motion as defined in classical mechanics in special relativity is Born rigidity. It maybe of use to you.

 

Also, what you suggest reminds me of light cones, which are invariant.

[Orodruin]

The Lorentz invariant hyper surfaces are by definition hyperboloids of the form (ct)^2 - x^2 - y^2 - z^2 = constant. Light cones are a special case of this where the constant is zero.

 

The hyperboloids have a special role as they may represent, for example, the mass-shell of a particle/object.

 

Furthermore, hyperbola such as (ct)^2 - x^2 = -k^2, y = z = 0, represent world-lines of observers with constant proper acceleration and their shape is also invariant under Lorentz transformations.

Edited April 27, 2014 by Orodruin