Elastic collisions

Lemma 12 Let in an elastic transversal collision between equivalent objects (see figure 6.2a) ¹ v, ¹ ·v1 ⇒ ¹ v, ¹−·v1 (6.2) reservoir particle ¹ ·v1 kick in from below with fixed velocity·v₁ and rebound antiparallel.

Then the incident object ¹ v moves on with same velocity v = R_αv deflected by an angle α

sin α

2

=

1−^v_c^x22

v · . (6.3)

10We start from the original physical grounds like Huygens, who did study the elastic collisions between identically constituted bodies to derive the collision laws for billiard balls, and like Einstein [46] and Feynman [49], who for the same reason investigate the interaction between bodies which collide and stick together.

Figure 6.2: c) symmetric elastic collision a) seen as elastic transversal collision when Alice drives by with same horizontal velocity to the left and b) when Bob drives to the right Proof: The elastic collision of identically constituted bodies¹ is well-defined by symmetry and Poincare covariance. One can freely adjust the scattering angle of an eccentric elastic collision (see figure 6.2c), e.g. with a suitable impact parameter between two billiard balls.

Let Alice and Bob fly by horizontally into opposite direction and observe the same process.

With Feynman’s trick [49] one can determine the kinematical description.

Let Alice drive by in a car with same horizontal velocity to the left. In her view the lower ball moves up and down vertically with the same velocity±·v₁(A)

w=

while the upper ball flies on with horizontal component u⁽_x^A) unvaried and a swap in the vertical component ∓u⁽_y^A) (we denote Alice intrinsic reference velocity v₁(A) and similarly Bob’s units v₁(B)). LetBob move relative to Alice with same horizontal velocity to the right

v_B =

u⁽_x^A) 0

·v₁(A) . (6.4)

Their measured values for duration, length transform by the Lorentz transformation t^(B) = t^(A)− _c^v2 ·x^(A)

From these basic quantities one induces the transformation for the derived quantity of ve-locity. One derives Bob’s velocity values for the colliding bodies

u⁽_x^B) := Δx^(B)

6.2. Basic measurement 121

From Bob’s perspective (on the same collision) the lower ball flies to the left with fixed horizontal velocity w⁽_x^B) and a reversion in the vertical component ±w⁽_y^B)

w=

w_x^(B)

±w_y^(B)

·v₁(B) u=

u⁽_x^B) u⁽_y^B)

·v₁(B) = 0

∓

·v₁(B)

while the upper ball moves up and down vertically with the same velocity·v₁(B). ForAlice andBob the roles of the upper and lower ball are reversed (see figure 6.2b). Alice andBob’s perspectives are equivalent; they observe equal ”vertical” impact velocities

u^(B) =^! −w^(A) = .

Finally Alice can express the vertical component of her ”horizontal” particle u⁽_y^{A) (6.4)(6}=^.7) ·

'

1− u⁽_x^A)2 c²

in terms of her impact velocity . Then the deflection angleα follows from figure 6.2a.

2 In the following construction outline we fix the transversal impact velocity·v₁of all reservoir elements. With given initial velocityv we can determine the deflection angleα(v, ) and vice versa, provided the latter we find the necessary initial velocityv(α, ).

Let one transversal standard kick (highlighted black in figure 6.3) deflect an incident particle ¹ v2 with velocity v₂ around angle α₂ = 18^◦; after a series of 9 more kicks of the same strength its direction is reversed; and similar for a slower particle ¹ v1 which requires half the standard kicks (highlighted purple in figure 6.3). In every transversal kick (6.2) we generate recoil particles ¹ ·v1 from an external reservoir withsame velocity·v₁ (depicted brown in figure 6.3). To capture and recycle all the temporarily mobilized steering elements from the center, we align all ”radial” standard kicks in the depicted way (pairwise along the dashed lines at diametrically opposed locations). In the total balance the reservoir particles do not appear. In the net result only the motion of the one particle ¹ v2 from bottom left and the (bundle of) two particles¹ v1,¹ v1 from the top right is reversed. We determine the relation between their admissible velocitiesv₂ andv₁ from matchingform andnumber of the radial kicksw_rad[¹ _v2,¹ _·v1] andw_rad[¹ _v1,¹ _·v1] (so that the total machinery functions).¹¹ By refinement of the building blocks one can construct similar models for the elastic collision between one particle¹ vn with high velocityv_n (from bottom left) and a spreading bundle of n standard particles ¹ _v1, . . . ,¹ _v1 (from top right). By construction (see figure 6.3) the incident bundle inherits an opening angle of orderα₁. In the refinement limit →0 (of an increasing number of radial kicks with negligible impact velocity ·v₁) the spreading of the bundle lim_→0α₁(v₁, )⁽⁶= 0 narrows: We get an elastic head-on collision (6.8).^.3)

11Every radial standard kick must deflect the incident particle¹ _v2 (highlighted black in figure 6.3) around one half of the deflection angleα2

=! ¹₂·α1than for the slower particle ¹ v1 (highlighted purple).

Figure 6.3: align standard collisions

6.2. Basic measurement 123

Proposition 5 Let one standard body¹ vn (from left) and a rigid composite ⁿ :=¹ ∗. . .∗ ¹ of n standard elements (from right) collide and repulse from one another with reversed velocities, symbolized

¹ vn, ⁿ v1 ⇒ ¹−vn, ⁿ−v1 . (6.8) Then in Poincare kinematics the initial velocities v_n, v₁ satisfy the relation

v_n

1− ^v_cⁿ2²

= −n· v₁

1−^v_c¹2²

. (6.9)

Proof: We approximate the elastic collision between two generic objects. Without restrict-ing generality let them be composites of unit objects ¹. We do not presuppose how the velocities change in more complex collisions. The trick is to mediate the direct interaction by an indirect replacement process with an external reservoir, which in the end is not ef-fected! Our model solely consists of elastic collisions between standard objects ¹ which must behave in a symmetrical way. We know the collision law for 1 + 1 equivalent objects by symmetry and relativity principle (Lemma 12). Based on it we construct the collision model for n+ 1 equivalent objects and ultimately for n+m composites (Corollary 5). We assemble intrinsically well-defined building blocks to a functioning configuration (criterium above). From the number and form we derive the amount of matter-velocity relation (6.9).

The construction outlined before follows the same steps as in Galilei kinematics [54]

except that the relativistic building blocks obey modified impact velocity-deflection angle relations v_n(α_n, ) and v₁(α₁, ) (6.3). As one can see from figure 6.3 we ultimately align the radial standard kicks (highlighted black resp. purple) against the fast incident object ¹ vn and against the n incident elements ¹ v1, . . . ,¹ v1 with matching deflection angles α₁ =^! n·α_n. In the refinement limit → 0 with α_i → 0, v_iy → 0, v_ix → v_i, sinα_i → α_i the matching condition is equivalent to sin(^α₂¹)= sin(n^! · ^α₂ⁿ)→n·sin(^α₂ⁿ) and implies with

Lemma 12

1− ^v_c¹2²

v₁ · ⁽⁶=^.3) n·

1−^v_c²ⁿ2

v_n ·

the relation (6.9) between the admissible initial velocities v₁,v_n. Our physical model medi-ates the elastic head-on collision between one unit object ¹ vn with velocity v_n(v₁, n) and a parallel beam ofnelements¹ v1, . . . ,¹ v1 with velocityv₁. Before and after the collision its elements fly with same velocityv₁ as if they were bound in a rigid composite ⁿ _v1.

2 Corollary 5 In an elastic head-on collision ¹ _vr, ^r_−v1 ⇒ ¹_−vr, ^r _v1 between unit object ¹ vr and a composite^r ofr:= ^m_n unit elements¹ v1 with standard velocityv₁ the admissible

velocity v_r satisfies

γ_vr·v_r = m

n ·γ_v1·v₁ .¹² (6.10)

Proof: Let the fragment^F play the role of acommon physical denominator of the standard body ¹ =: ^F ∗. . . ∗ ^F express the head-on collision between the standard body ¹ vr and the (rational) composite ^r _v1 in terms of their common fragments: the composite n · ^F _vr of n fragments with admissible velocityv_r from left runs into the other compositem· ^F v1 of m fragments with standard velocity v₁ from right (see figure 6.4a).

Let Alice have access to an external ”fragment reservoir”

Sf

Alice generatesm·n anti-parallel fragment-pairs. We chose fragment velocity v_f(v₁, n) γ_v1·v₁ ⁽⁶=^.9) n·γ_v

f ·v_f (6.11)

such that in an elastic head-on collision n· ^F _vf, ^F _−v1 ⇒^wn n· ^F _−vf, ^F _v1 the composite n· ^F of n equivalent elements ^F vf with fragment velocity v_f rebounds antiparallel from one single fragment ^F v1 with standard velocity v₁ (see figure 6.4b). Further we fix the admissible velocity v_r(v_f, m)

γ_vr ·v_r ⁽⁶=^.9) m·γ_v

f ·v_f (6.12)

such that in an elastic head-on collision ^F _vr, m· ^F_−vf ^w⇒ ^{m F} _−vr, m· ^F _vf one single fragment ^F vr with initial velocity v_r rebounds antiparallel from the composite m· ^F of m equivalent elements ^F v1 with fragment velocity v_f.

The fragment pairs

^F −vf,^F vf

mediate an elastic head-on collision between an inci-dent composite n· ^F _vr from left and another composite m· ^F _−v1 from right

Each fragment rebounds antiparallel with same velocity. Alice can recycle (m·n)×w_f⁻¹ all temporarily expended resources back into the fragment reservoir

Sf

0,^F 0

(see figure

12In our calorimeter model {6.2.2} we will process fragments of standard impulse carriers ¹ v1. We abbreviate the common term ¹

1−^v_c2²

=:γ in all arithmetic expressions.

13By our pre-theoretic construction principle identically constituted bodies behave symmetrical in elastic head-on collisions. The size of the standard object¹ is variable or composable from smaller fragments ^F .

6.2. Basic measurement 125

Figure 6.4: a) The direct elastic head-on collision of two composites is b) mediated by temporarily activated fragment pairs ^F _−vf, ^F _vf. In the end c) all steering resources are recycled back into the fragment reservoir

S_f

v=0,^F _v=0 .

6.4c). The reservoir mediated collision has the same initial and final state as the direct elastic collision of both composites. The admissible initial velocityv_r(v₁,^m_n) is given from

γ_vr ·v_r ^(6.12)(6=^.11) m

n ·γ_v1 ·v₁ .

2

Im Dokument Physical determination of the action (Seite 127-134)