4vec-v-p-eCalculation < Main

*Velocity 4-vector*

define the u_4 vector note the differentiation with respect to ds.

u=(u_t, u_x, u_y, u_z) = (cdt/ds, dx/ds, dy/ds, dz/ds)  note that by its definition the 4-velocity is dimensionless like beta=v/c

take the u^2 = u_t^2-u_x^2-u_y^2-u_z^2 = [ (cdt)^2- dx^2-dy^2-dz^2]/ ds^2 = ds^2/ds^2 = 1 

express ds^2=gamma^(-2) c^2 dt^2 =&gt; (c^2dt^2)/(ds^2)= gamma^(2) 

then the u_t component above is cdt/ds= sqrt(c^2dt^2/ds^2) = sqrt (gamma^2) ==&gt; u_t= gamma

__everybody should reproduce the above__

and then u_4 =( gamma, v/c gamma) 

*Momentum 4-vector*

newtonian *P* =m<strong>v</strong> where v=3-velocity  

Define 4-momentum as follows 

P_j= m_0 * u_j * c where u_j is the the famous 4-velocity (i.e. j runs 0(time) , 1(x), 2(y), 3(z)) 

 then the 3-momentum using the 3-velocity (v_a) can be expressed as 

P_a=m_0 * u_a * c = m_0 * (gamma v_a/c ) * c  (taking the 3-velocity from above)

               = m_0 * gamma * v_a this should be equal to <strong>P</strong>=m<strong>v</strong> (newton) because it is the 3-component part of the 4-vector ==&gt; <strong>m= </strong> *m_0 * gamma* <strong> (this says that the total mass of the moving particle is bigger than its rest mass m_0)</strong>

the temporal component of the momentum 4-vector as we defined above is 

P_t= m_0 * u_t * c = *m_0 * gamma ** c = <strong>m*c </strong>

from the Taylor expanded mass we had m*c^2=m_0*c^2+KE and we said the total energy of the moving particle is

 E_tot=m*c^2 = *m* c* * c = P_t *c =&gt; P_t= E_tot/c ==&gt; 

 the temporal component of the momentum is the total energy divided by c

So now we look for the direct relation between E_tot, and P_j 

P_j = (P_t, <strong>P) ==&gt; </strong> *P_j^2* <strong>= P_t^2- P^2 = (E_tot/c)^2- P^2 ==&gt; </strong>

*(m_0 * u_j* c)^2* <strong>= (E_tot/c)^2- P^2 ==&gt; (u_j^2=1 from the 4-velocity)</strong>

    *m_0^2 *c^2 = E_tot^2/c^2 - P^2      ==&gt;*

               *E_tot= +- sqrt(m0^2*c^4+P^2c^2)* 

-- Main.smaria - 2011-01-21

This topic: Main > Smaria > 4vec-v-p-eCalculation
Topic revision: r1 - 2011-01-21 - smaria

Copyright © by the contributing authors. All material on this collaboration platform is the property of the contributing authors.
Ideas, requests, problems regarding TWiki? Send feedback