Summary: Continuing notes on the tensor version of the Lorentz tranform. It's time to start on the second set of examples.
The interval in four space is invariant under Lorentz transforms and is called the Lorentz scalar.
The Lorentz transform also applies to differential distances as,
dx′μ=Λμνxμ
We were asked in class to work out x2+y2+z2−t2=x′2+y′2+z′2−t′2
The transforms we'll use are:
x=γ(x′+vt′)
t=γ(t′+vx′)
Substituting these into the l.h.s. gives
γ2(x′+vt′)2−γ2(t′+vx′)2=x′2−t′2
=γ2(x′2+2vtx+v2t′2)−γ2(t′2+2vxt+v2x′2)=x′2−t′2
=γ2(x′2+v2t′2)−γ2(t′2+v2x′2)=x′2−t′2
γ2x′2−γ2v2x′2=γ2x′2(1−v2)=x′2
Similarly
γ2v′2t′2−γ2t′2=−γ2t′2(−v2+1)=−t′2
So
x′2−t′2=x′2−t′2
Done
The interval in four space is invariant under Lorentz transforms and is called the Lorentz scalar.
The Lorentz transform also applies to differential distances as,
dx′μ=Λμνxμ
We were asked in class to work out x2+y2+z2−t2=x′2+y′2+z′2−t′2
The transforms we'll use are:
x=γ(x′+vt′)
t=γ(t′+vx′)
Substituting these into the l.h.s. gives
γ2(x′+vt′)2−γ2(t′+vx′)2=x′2−t′2
=γ2(x′2+2vtx+v2t′2)−γ2(t′2+2vxt+v2x′2)=x′2−t′2
=γ2(x′2+v2t′2)−γ2(t′2+v2x′2)=x′2−t′2
γ2x′2−γ2v2x′2=γ2x′2(1−v2)=x′2
Similarly
γ2v′2t′2−γ2t′2=−γ2t′2(−v2+1)=−t′2
So
x′2−t′2=x′2−t′2
Done
Comments
Post a Comment
Please leave your comments on this topic: