4-force, 4-momentum, energy and mass relations.

[heitor]

This is an exercise of Special Relativity the professor asked last week.
Sorry for the long post, I hope you don't get bored reading it, also, this is my first post here :shy:

Homework Statement

Defining the 4-force that acts on a particle as the proper-time variation of the 4-momentum F^\mu := \frac{d P^\mu}{d \tau}.

1. Justify Einsteins relation between mass and energy: E = mc^2
2. Show, using the \eta _{\mu\nu}P^\mu P^\nu, that E^2 = (mc^2)^2 + (\vec{p}^2c^2), where \vec{p} is the 3-momentum.
3. Show that in SR the 3-force and 3-acceleration is not always parallel to each other.

Homework Equations

I'm using the convention: x^0 = ct and (- + + +) for the metric tensor.

4-velocity: u^\mu = \gamma_{(v)} (c, \vec{v}), with \vec{v} the 3-velocity in lab frame.

And some results I got, they are somewhere in Wikipedia and some books also.

gamma: \frac{d}{d \tau} = \frac{d t}{d \tau} \frac{d}{d t} = \gamma_{(v)} \frac{d}{d t}

I'll omit (v) in gamma for brevity.

derivative of gamma: \dot{\gamma} = \frac{d \gamma}{dt} = \gamma ^3 \frac{\vec{v} \cdot \vec{a}}{c^2}

acceleration: a^\mu = \left ( \gamma ^4 \frac{\vec{v} \cdot \vec{a}}{c}, \gamma^4 \vec{a}_\parallel + \gamma ^2 \vec{a}_\perp \right )

where \vec{a}_\parallel = (\vec{a} \cdot \vec{v}) \vec{v}/v^2 is the parallel component of the 3-acceleration to the 3-velocity and \vec{a}_\perp = \vec{a} - \vec{a}_\parallel is the perpendicular one.​

3. The Attempt at a Solution for question 1 and 2

I think I got the results, but i have some doubts.

F^\mu = \frac{d P^\mu}{d \tau} = \left(\frac{d}{d \tau}(m \gamma c), \frac{d}{d \tau}(m \gamma \vec{v}) \right )
I'll call this 'result' as (EQ1).

Using the gamma relation in the relevant equations:
F^\mu = \gamma \left ( m c \dot{\gamma}, \frac{d\vec{p}}{dt} \right ) = \left ( m c \gamma ^4 \frac{\vec{v} \cdot \vec{a}}{c^2}, \gamma \vec{f}_R \right )
I'll call this 'result' as (EQ2). Where \vec{f}_R is the 3-force. We know that F^\mu is mass times acceleration:

F^\mu = m a^\mu = m \left ( \gamma ^4 \frac{\vec{v} \cdot \vec{a}}{c}, \gamma^4 \vec{a}_\parallel + \gamma ^2 \vec{a}_\perp \right )
This is (EQ3).

Comparing (EQ2) and (EQ3) we get:
\vec{f}_R = m \left ( \gamma^3 \vec{a}_\parallel + \gamma \vec{a}_\perp \right )

scalar product with \vec{v} gives a variation in energy:
\vec{f}_R \cdot \vec{v} = m \gamma^3 \vec{a} \cdot \vec{v} = \frac{dE}{d \tau}
This is (EQ4).

Here is the derivative with respect to the proper-time, right? I'm not sure about this...​

Putting (EQ4) inside (EQ3) in the first component and re-using the second component of (EQ2):
F^\mu = \gamma \left (\frac{\vec{f}_R \cdot \vec{v}}{c}, \vec{f}_R \right ) = \left ( \frac{1}{c} \frac{d E}{d\tau}, \frac{d\vec{p}}{d\tau} \right )
This is (EQ5).

Comparing (EQ5) and (EQ1) yelds:
\frac{dE}{d\tau} = \frac{d}{d\tau} (m \gamma c^2)

So:

E = m \gamma c^2 +\ constant
This is (EQ6).

This is almost what the question 1 asks, but what is this constant? What is the meaning of E? Is it kinect + 'rest energy'?​

Question 2 I got:

\eta _{\mu\nu}P^\mu P^\nu = m^2 \gamma ^2 (-c^2 + \vec{v}^2) = -m^2 c^2
(EQ7)

From (EQ6) we can write the 4-momentum as:
P^\mu = \left ( \frac{E}{c}, \vec{p} \right )
(EQ8)

so:
\eta _{\mu\nu}P^\mu P^\nu = - \frac{E^2}{c^2} + \vec{p}^2 = -m^2 c^2

which is the answer to question 2 if the constant of (EQ6) is zero...



4. The attempt at a solution for question 3

If we do the same trick used to split the acceleration in parallel and perpendicular parts:

\vec{f}_R := \vec{f}_{R, \parallel} + \vec{f}_{R, \perp}
We build the parallel one:

\vec{f}_{R, \parallel} = \frac{(\vec{f}_R \cdot \vec{a}) \vec{a}}{\vec{a}^2}
and the perpendicular one:

\vec{f}_{R, \perp} = \vec{f}_R - \vec{f}_{R, \parallel} = \vec{f}_R - \frac{(\vec{f}_R \cdot \vec{a}) \vec{a}}{\vec{a}^2}

But I was unable to prove \vec{f}_{R, \perp} \neq \vec{0}.

Do you have any hint? All that I got was some previous equations.


If I was not clear in some statement, please tell me.

(is there a way to put a 'name' in some equations to be displayed on the right side of it? Like latex documents?)
(Sorry for my bad English...)

Thanks in advance,
Heitor.

 

[dauto]

From the expression \vec{f}_R = m \left ( \gamma^3 \vec{a}_\parallel + \gamma \vec{a}_\perp \right ) we see that the force is not parallel to the acceleration because \gamma^3 is different from \gamma so the parallel and perpendicular components scale differently.

 

[dauto]

By the way, I think that instead of taking equations from books and wikipedia, you ought to prove them.

 

[heitor]

From the expression \vec{f}_R = m \left ( \gamma^3 \vec{a}_\parallel + \gamma \vec{a}_\perp \right ) we see that the force is not parallel to the acceleration because \gamma^3 is different from \gamma so the parallel and perpendicular components scale differently.

Is there any interpretation for this? A 3-force in Special Relativity does not have the same meaning as in Newtonian Mechanics?


I did proved them, it was the previous question in the list.

 

[dauto]

The 3-force is derived from force laws just like in Newtonian mechanics but f=ma doesn't apply. instead we have f = m (γ³a_parallel + γa_{perpendicular})