String Theory: Light-cone Coordinates




Special Relativity:

Spacetime Coordinates

Eqn 2.7: \(x^{\mu}\) = \((x^{0}, x^{1}, x^{2}, x^{3})\) \( \equiv \) \( (ct, x, y, z) \)

(Time coordinate scaled by c, so all coordinates have units of length.)


Lorentz Frames

Lorentz frame S: Two events represented by coordinates \(x^{\mu}\) and (\(x^{\mu}\) + \(\Delta x^{\mu} \))
Lorentz frame S': Two events represented by coordinates \(x'^{\mu}\) and (\(x'^{\mu}\) + \(\Delta x'^{\mu} \))

In general, x and x' are different, and so are the coordinate differences \(\Delta x\) and \(\Delta x'\).

Invariant Interval \(\Delta s^{2}\)

However, observers will agree on the value of \( \Delta s^{2} \). (We do not square root it, because it can be negative)

- \(\Delta s^{2}\) = - ( \( \Delta x^{0})^{2}\) + (\( \Delta x^{1})^{2}\) + (\( \Delta x^{2})^{2}\) + (\( \Delta x^{3})^{2}\)
= - (\( \Delta x'^{0})^{2}\) + (\( \Delta x'^{1})^{2}\) + (\( \Delta x'^{2})^{2}\) + (\( \Delta x'^{3})^{2}\)
= - \(\Delta s'^{2}\)

Minus sign implies \( \Delta s^{2} \)> 0 for timelike separated events.

Separation Types

Timelike separated: \(\Delta s^{2}\) > 0
Lightlike separated: \(\Delta s^{2}\) = 0
Spacelike separated: \(\Delta s^{2}\) < 0

Differentials

Re-define the time coordinate to bury the minus sign:

\(dx_{0}\) \( \equiv \) - \(dx^{0}\) , \(dx_{1}\) \( \equiv \) \(dx^{1}\) , \(dx_{2}\) \( \equiv \) \(dx^{2}\) , \(dx_{3}\) \( \equiv \) \(dx^{3}\)
\( \Rightarrow dx_{\mu}\) = (\(dx_{0}, dx_{1}, dx_{2}, dx_{3})\)

Einstein Summation Convention

Expressing \( ds^{2} \) in lower and upper case:

-\(ds^{2}\) = \(dx_{0} dx^{0} + dx_{1} dx^{1} + dx_{2} dx^{2} + dx_{3} dx^{3}\) = \(\underset{\mu}{\Sigma} dx_{\mu} dx^{\mu}\)



Light-cone Coordinates

In the Barton Zwiebach book, relativistic strings are quantized in a special coordinate system, called light-cone coordinates. Lorentz covariant quantization involves no special coordinates, but mostly goes beyond the scope of this book. Light-cone coordinates \(x^{+}\) and \(x^{-}\) are two independent linear combinations of the time coordinate and one of the spatial coordinates.

The light-cone coordinates are the associated coordinate axes of the world-lines of light from the origin along the \(x^{1}\) axis:

\(x^{+} \equiv \frac{1}{\sqrt{2}} (x^{0} + x^{1}) \) \(x^{-} \equiv \frac{1}{\sqrt{2}} (x^{0} - x^{1}) \)

The light-cone coordinate system is then: \( (x^{+}, x^{-}, x^{2}, x^{3}) \)

It follows that if a light beam moves in the positive \( x^{1} \) direction, \( x^{1} = ct = x^{0} \), therefore \( x^{-} = 0 \). If the light beam moves in the negative \(x^{1} \) direction, \( x^{1} = -ct = x^{0} \), therefore \( x^{+} = 0 \). This is a 45 degree angle with the \(x^{0} \) and \(x^{1} \) axes. Since the light-cone coordinates are in a sense time coordinates, time freezes for light beams. \( x^{+} \) will be taken as the "light-cone time" coordinate and \( x^{-} \) will be treated as a spatial coordinate.

The invariant interval is:

- \( ds^{2} = - 2 dx^{+} dx^{-} + (dx^{2})^{2} + (dx^{3})^{2} \)

This is because:

\( dx^{+} dx^{-} \) = \( (\frac{1}{\sqrt{2}})^2 \) [\( (dx^{0} + dx^{1})(dx^{0} - dx^{1}) \)] = \( \frac{1}{2} \) [ \( (dx^{0})^{2} - (dx^{1})^{2} \)]

The ability to solve for \( dx^{+} \) or \( dx^{-} \) from \( ds^{2} \) without doing a square root is an important feature.



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Last updated: 8/2/2020