Relativistic Kinematics: Invariants and Conservation Laws

A particle accelerator creates proton-antiproton pairs by bombarinding a stationary proton target with an accelerated proton. What is the threshold energy in the laboratory of the accelerated proton that is just necessary to create a single proton-antiproton pair in addition to the original two protons? (Protons and anti-protons have the same rest mass.)

QUESTIONS
(Score is number right minus number wrong.)

CONSERVATION LAWS tell us that certain quantities are the same at different TIMES. INVARIANTS tell us that certain quantities are the same in different FRAMES.

The 4-momentum inner product of the single accelerated proton before collision in terms of quantities measured in the laboratory frame is given by

: Equation (3a).; Equation (3b).; Equation (3c).; Equation (3d).

The 4-momentum inner product of the single accelerated proton before collision taken in its own rest frame is given by

: Equation (4a).; Equation (4b).; Equation (4c).; Equation (4d).

The 4-momentum inner product of the single accelerated proton before collision taken in BOTH the rest frame of the proton AND the laboratory frame, no matter what energy and momentum it has in the laboratory frame, is always correctly given by given by

: Equation (4a).; Equation (4b).; Equation (4c).; Equation (4d).; None of these.

The inner product (Equation (2)) is an invariant for the 4-momentum of each of the two particles in FRAME (1) (upper, left, see diagram) taken separately but the inner product is not an invariant for the total 4-momentum of the two particles taken together.

: True; False

The TOTAL 4-momentum of both protons taken together in the laboratory (FRAME (1) in the diagram) in terms of quantities measured in the laboratory frame is given by (quantities referring to the accelerated proton only carry a subscript (1))

: Equation (5a); Equation (5b); Equation (5c); Equation (5d)

The total 4-momentum of the 3 protons + 1 antiproton in the laboratory (FRAME (3)) in terms of quantities measured in the laboratory is given by

: Equation (6a); Equation (6b); Equation (6c); Equation (6d)

The values of gamma and momentum as they appear in Equations (5) and (6) are correctly related as shown in Equation (7).

: True; False

The total 4-momentum of the 3 protons + 1 antiproton in the center-of-momentum frame (FRAME (4)) in terms of quantities measured in the laboratory is given by

: Equation (6a); Equation (6b); Equation (6c); Equation (6d)

Using the notation that (i=1,4) refers to the frames labeled in the diagram, which of the following is true for the total 4-momentum?

: Equation (8a).; Equation (8b).; Equation (8c).; Equation (8d).; Equation (8e).

Using the notation that (i=1,4) refers to the frames labeled in the diagram, which of the following is true for the total 4-momentum?

: Equation (9a).; Equation (9b).; Equation (9c).; All of these.; None of these.

Using the notation that (i=1,4) refers to the frames labeled in the diagram, which of the following is true?

: Equation (10a).; Equation (10b).; Equation (10c).; All of these.; None of these.

Applying Equation (9), the threshold energy for the accelerated proton in the laboratory to just produce a proton-antiproton pair is given by

: Equation (11a).; Equation (11b).; Equation (11c).; Equation (11d).

Now build a new accelerator (a "collider") that accelerates two protons to the same threshold energy needed by the one proton when a stationary target is used. Collide them so that the laboratory frame FRAME (1) and the center-of-momentum frame (FRAME (2) become the same frame (as do FRAMES (3) and (4)).

By what factor has the total amount of energy available for making particles in the center-of momentum frame increased by replacing the original fixed-target accelerator with the collider?

: 8/4 = 2 .; 10/4 = 2.5 .; 12/4 = 3.0 .; 14/4 = 3.5 .; 16/4 = 4.0 .

EQUATIONS

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