Energy and momentum in special relativity

G. N.

 

Lewis

and

R. C.

 

Tolman

, “,”  , – ().

R. D.

 

Carmichael

, “,”  , – ().

R. P.

 

Feynman

,

R. B.

 

Leighton

, and

M.

 

Sands

, (, , ), Vol. , Sec. 16–4.

E. F.

 

Taylor

and

J. A.

 

Wheeler

, (, , ), Chap. 2.

A. P.

 

French

, (, , ), Chap. 6.

N.

 

Ashby

and

S. C.

 

Miller

, (, , ), Sec. 4.3.

E. E.

 

Anderson

, (, , ), Chap. 4, Sec. 1.

F. J.

 

Blatt

, (, , ), Sec. 2.2.

A.

 

Beiser

, , 5th ed. (, , ), Sec. 1.7.

P. A.

 

Tipler

and

R. A.

 

Llewellyn

, , 4th ed. (, , ), Sec. 2.1.

D.

 

Giulini

, Special Relativity: A First Encounter, $100Years$ Since Einstein (, , ), Sec. 3.8.

S. T.

 

Thornton

and

A.

 

Rex

, , 3rd ed. (, , ), Sec. 2.11.

Additional references can be found in the first endnote of

P. C.

 

Peters

, “,”  , – ().

Symmetry and the principle of relativity constrain the velocity of the final state particle $M$⁠. If $M$ had a nonvanishing $y$ velocity, it would move either in the direction of the particle launched vertically in frame $S$ (if $Vy<0$⁠), or in the direction of the particle launched vertically in frame $S′$ (if $Vy>0$⁠). A nonzero value of $Vy$ would indicate the presence of a preferred frame, either $S$ or $S′$⁠, contradicting the principle of relativity.

The equality of $∣Vx∣$ and $∣Vx′∣$ in Fig. 1 is also a consequence of symmetry and the principle of relativity, because if $∣Vx∣≠∣Vx′∣$⁠, say $∣Vx∣>∣Vx′∣$⁠, then we could distinguish frame $S$ from frame $S′$ in a way that would violate the principle of relativity. The velocity transformation Eq. (6a) with $v∥=Vx=U$ and $v∥′=Vx′=−U$ gives Eq. (7).

Our derivation of Eq. (10) is valid only for $w>0$ and not for $w=0$⁠, so we cannot set $w=0$ in Eq. (10). This point was made by

W. L.

 

Kennedy

, “,”  , – ().

R. C.

 

Tolman

, “,”  , – ().

M.

 

Born

, (, , ), Chap. 6.

J. G.

 

Taylor

, (, , ), Chap. 5.

L. C.

 

Baird

, “,”  , ().

P. D.

 

Gupta

, “,”  , ().

S.

 

Adams

, (, , ), Sec. 2.12.

In a longitudinal collision all particles move along a line, say the $x$ axis. Momentum conservation implies that $∑imivixf(vi)=∑fmfvfxf(vf)$⁠, where $mi$ is the mass and $v⃗i$ the velocity of initial particle $i$⁠, and $mf$ is the mass and $v⃗f$ the velocity of final particle $f$⁠. Consider a new frame moving in the $−y$ direction with speed $w$⁠. In this frame $y$ momentum conservation implies that $∑imiwf(w2+(vi∕γ(w))2)=∑fmfwf(w2+(vf∕γ(w))2)$⁠. This relation holds for any $w$⁠. In the limit $w→0$ we find $∑imif(vi)=∑fmff(vf)$⁠, which can be interpreted as conservation of relativistic mass. See endnote 2 in Ref. 13 and p. in Ref. 4.

C. G.

 

Adler

, “,”  , – ().

L. B.

 

Okun

, “,”  (), – ().

R.

 

Baierlein

, “,”  , – ().

T. R.

 

Sandin

, “,”  , – ().

R.

 

Penrose

and

W.

 

Rindler

, “,”  , – ().

J.

 

Ehlers

,

W.

 

Rindler

, and

R.

 

Penrose

, “,”  , – ().

M. J.

 

Feigenbaum

and

N. D.

 

Mermin

, “,”  , – ().