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Ch. 5 Problems

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Basic

Medium

Basic

B-1. Metric in Light-Cone Coordinates

From the definition of light-cone coordinates \(x^\pm = (x^0 \pm x^1)/\sqrt{2}\), show by direct calculation that

\[ ds^2 = -(dx^0)^2 + (dx^1)^2 + (dx^2)^2 + (dx^3)^2 \]

can be rewritten as

\[ ds^2 = -2\,dx^+ dx^- + (dx^2)^2 + (dx^3)^2 \]

(with \(c = 1\)).

Hint

Substitute \(dx^0 = (dx^+ + dx^-)/\sqrt{2}\), \(dx^1 = (dx^+ - dx^-)/\sqrt{2}\), or alternatively use \(dx^+ dx^- = \frac{1}{2}[(dx^0)^2 - (dx^1)^2]\).

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B-2. Matrix Representation of the Light-Cone Metric

Find the inverse matrix \(\hat{\eta}^{\mu\nu}\) of the light-cone metric matrix (with row and column ordering \(+, -, 2, 3\))

\[ \hat{\eta}_{\mu\nu} = \begin{pmatrix} 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

and verify that it satisfies \(\hat{\eta}^{\mu\lambda}\hat{\eta}_{\lambda\nu} = \delta^\mu{}_\nu\).

Hint

The \((+,-)\) block is simply the \(2 \times 2\) anti-diagonal matrix \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\), so its inverse has the same form.

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B-3. Four-Momentum in Light-Cone Coordinates and \(p^-\)

Express the four-momentum \(p^\mu = (p^0, p^1, p^2, p^3)\) of a particle with mass \(m\) in terms of light-cone coordinate components \(p^\pm = (p^0 \pm p^1)/\sqrt{2}\), \(p^2, p^3\).

(a) Write the invariant norm \(p^\mu p_\mu\) of the four-momentum in light-cone coordinates. Show that the result is \(-2 p^+ p^- + (p^2)^2 + (p^3)^2\).

(b) Using \(p^\mu p_\mu = -m^2\), solve for \(p^-\) in terms of \(p^+, p^2, p^3, m\) and derive

\[ p^- = \frac{(p^2)^2 + (p^3)^2 + m^2}{2\,p^+} \]

(c) In ordinary coordinates, solving \(p^\mu p_\mu = -m^2\) for \(p^0\) gives \(p^0 = \pm\sqrt{|\vec{p}|^2 + m^2}\), leaving a \(\pm\) sign ambiguity. In light-cone coordinates, \(p^-\) is uniquely determined as shown in (b). Explain this difference physically.

Hint

(a) Use \(\eta_{\mu\nu}p^\mu p^\nu = -(p^0)^2 + (p^1)^2 + (p^2)^2 + (p^3)^2\) together with \((p^0)^2 - (p^1)^2 = 2 p^+ p^-\). (c) Consider how the "sign of energy" and the "distinction between positive/negative energy solutions" are handled in light-cone coordinates.

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Medium

M-1. Inner Product in Light-Cone Coordinates

Express the Minkowski inner product \(A^\mu B_\mu = \eta_{\mu\nu}A^\mu B^\nu\) of two 4-vectors \(A^\mu, B^\mu\) in terms of the light-cone coordinate components \(A^\pm, A^2, A^3\) and \(B^\pm, B^2, B^3\).

Hint

Rewrite \(A^\mu B_\mu = -A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3\).

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M-2. Lorentz Transformation (Boost) in Light-Cone Coordinates

A boost in the \(x^1\) direction (where the \(S'\) frame moves with velocity \(v\) in the \(x^1\) direction relative to the \(S\) frame) can be written using the rapidity \(\varphi\) (\(\tanh\varphi = v\)) as

\[ \begin{pmatrix} x^{0'} \\ x^{1'} \end{pmatrix} = \begin{pmatrix} \cosh\varphi & -\sinh\varphi \\ -\sinh\varphi & \cosh\varphi \end{pmatrix} \begin{pmatrix} x^0 \\ x^1 \end{pmatrix} \]

(see General Relativity 3.3 "Derivation of the Lorentz Transformation"). Rewrite this in light-cone coordinates \(x^\pm\) and show the following.

(a) \(x^{+\prime} = e^{-\varphi}\,x^+\), \(x^{-\prime} = e^{\varphi}\,x^-\) (a boost in the \(x^+\) direction acts as a scale transformation in light-cone coordinates).

(b) Therefore, the product \(x^+ x^-\) is invariant. Verify that this is consistent with the invariance of the \(-2\,dx^+ dx^-\) part of \(ds^2\).

Hint

(a) Use \(\cosh\varphi \pm \sinh\varphi = e^{\pm\varphi}\). (b) \((e^{-\varphi}dx^+)(e^\varphi dx^-) = dx^+ dx^-\).

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