Suppose a particle, say a positron is moving with initial velocity ${\bf v} = (c/3,c/3,c/3)$, where $c$ is the speed of light. Then the relativistic momentum is ${\bf p} = \gamma m{\bf v}$. Now suppose we apply a force, perhaps due to an electric field E =(0,0,E), so the force is F= (0,0,eE). In Newtonian mechanics this force can only change the z component the momentum and the x and y components are conserved as no force acts in their direction.
In the relativistic case I don't understand how the components of the momentum will change as they cannot be independent of each other. Here is my frame of thought. An increase in the z component of the velocity will increase the $\gamma$ factor. But if the momentum in the x and y direction is conserved then the x and y components of the velocity must decrease to compensate for an increasing $\gamma$ factor.
Is this reasoning the physical reality?

Suppose a particle, say a positron is moving with initial velocity ${\bf v} = (c/3,c/3,c/3)$, where $c$ is the speed of light. Then the relativistic momentum is ${\bf p} = \gamma m{\bf v}$. Now suppose we apply a force, perhaps due to an electric field E $=(0,0,E)$, so the force is F$= (0,0,eE)$. In Newtonian mechanics this force can only change the z component the momentum and the x and y components are conserved as no force acts in their direction.
In the relativistic case I don't understand how the components of the momentum will change as they cannot be independent of each other. Here is my frame of thought. An increase in the $z$ component of the velocity will increase the $\gamma$ factor. But if the momentum in the $x$ and $y$ direction is conserved then the $x$ and $y$ components of the velocity must decrease to compensate for an increasing $\gamma$ factor.
Is this reasoning the physical reality?

Suppose a particle, say a positron is moving with initial velocity v = (c/3,c/3,c/3), where c is the speed of light. Then the relativistic momentum is p = $\gamma$ mv. Now suppose we apply a force, perhaps due to an electric field E =(0,0,E), so the force is F= (0,0,eE). In Newtonian mechanics this force can only change the z component the momentum and the x and y components are conserved as no force acts in their direction.
In the relativistic case I don't understand how the components of the momentum will change as they cannot be independent of each other. Here is my frame of thought. An increase in the z component of the velocity will increase the $\gamma$ factor. But if the momentum in the x and y direction is conserved then the x and y components of the velocity must decrease to compensate for an increasing $\gamma$ factor.
Is this reasoning the physical reality?

Suppose a particle, say a positron is moving with initial velocity ${\bf v} = (c/3,c/3,c/3)$, where $c$ is the speed of light. Then the relativistic momentum is ${\bf p} = \gamma m{\bf v}$. Now suppose we apply a force, perhaps due to an electric field E =(0,0,E), so the force is F= (0,0,eE). In Newtonian mechanics this force can only change the z component the momentum and the x and y components are conserved as no force acts in their direction.
In the relativistic case I don't understand how the components of the momentum will change as they cannot be independent of each other. Here is my frame of thought. An increase in the z component of the velocity will increase the $\gamma$ factor. But if the momentum in the x and y direction is conserved then the x and y components of the velocity must decrease to compensate for an increasing $\gamma$ factor.
Is this reasoning the physical reality?