Centre of mass (in relativity).

In summary, the collision between a bullet of rest mass M and a stationary target of rest mass m results in a post-collision gamma factor of M^2+M^2)/(2mM) that cannot exceed (m^2+M^2)/(2mM).
  • #1
MathematicalPhysicist
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consider a head on elastic collision of a bullet of rest mass M with a stationary target of rest mass m.
prove that the post-collision gamma factor of the bullet cannot exceed (m^2+M^2)/(2mM)

im given as a hint to prove that if P,P' are the pre and post 4 momentum of the bullet and Q,Q' are of the target, then in the CM frame (P'-Q)^2>=0
well i understand how from the hint i get the answer, but how to arrive at this, i mean the momentum is conserved, i.e: P'+Q'=P+Q
so by interchanging we get:
P'-Q=P-Q'
now in the cm, we must have: (P'-Q)^2-(P-Q')^2=(M+m)^2c^2
but i don't quite see why this is correct, i mean
P=(E/c,p)
Q=(mc,0)
P'=(E'/c,p')
Q'=(E''/c,p'')
where in the cm we have: p'=-p''
so (cause the 3 momentum there is zero) (P'-Q)^2=(E'/c)^2+(mc)^2-2*E'*m
(P-Q')^2=(E/c)^2+(E''/c)^2-2EE''/c^2
here's where I am stuck, can anyone help?
 
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  • #2
Are you asking why the square of a real number must be non-negative? :confused: Maybe I'm misunderstanding...
 
  • #3
your'e correct, i thought about it myself, and there's not a lot to prove here, but then why is the hint to use the cm system to prove that (P'-Q)^2>=0?
 
  • #4
Gokul43201 said:
Are you asking why the square of a real number must be non-negative? :confused: Maybe I'm misunderstanding...

P and P' are four-vectors, so I assume that (P - P')^2 is actually
[itex](P - P')_0^2 - (P - P')_1^2 - (P - P')_2^2 - (P - P')_3^2[/tex]
where the [itex]P_i[/itex] are real numbers, in which case it is not trivial at all that the "square" is positive.
 
  • #5
so how would you go around proving it?
my exam is tomorrow... (-:
 
  • #6
CompuChip said:
P and P' are four-vectors, so I assume that (P - P')^2 is actually
[itex](P - P')_0^2 - (P - P')_1^2 - (P - P')_2^2 - (P - P')_3^2[/tex]
where the [itex]P_i[/itex] are real numbers, in which case it is not trivial at all that the "square" is positive.
Next time, I'll actually read the question properly before I shoot my mouth... err, keyboard off!
 
  • #7
I'm a bit confused by all the primes, I think you still need that
(E/c)^2+(E''/c)^2-2EE''/c^2 >= 0.
Can't you use some estimate like E > E'', which would give
(E/c)^2+(E''/c)^2-2EE''/c^2 >= (E/c)^2+(E/c)^2-2E^2/c^2 = 0
where the estimate is justified by a physical argument?
 

What is the centre of mass in relativity?

The centre of mass in relativity refers to the point where the mass of a system is concentrated or can be considered to be concentrated. It is a mathematical concept used to describe the motion of objects in relativity.

How is the centre of mass calculated in relativity?

In relativity, the centre of mass is calculated using the equations for special relativity, specifically the Lorentz transformation equations. These equations take into account the relative velocities and positions of the objects in the system to determine the centre of mass.

Why is the centre of mass important in relativity?

The centre of mass is important in relativity because it is a frame-independent quantity, meaning that it remains the same regardless of the observer's frame of reference. This allows for a consistent and accurate description of the system's motion in different frames of reference.

Can the centre of mass move faster than the speed of light?

No, according to the principles of special relativity, the speed of light is the maximum speed at which any object or information can travel. Therefore, the centre of mass cannot move faster than the speed of light.

How does the centre of mass relate to gravity in relativity?

In general relativity, gravity is described as the curvature of spacetime caused by the presence of mass and energy. The centre of mass is used to determine the overall gravitational effects of a system, as it represents the point where the mass is concentrated. It is also used in calculations for the gravitational force between two objects.

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