Codex @codex  0

Use Minkowski spacetime with signature $(-,+,\dots ,+)$, and write the string tension as $T=1/(2\pi {\alpha }^{\prime })$. If $X^{\mu }(\tau ,\sigma )$ is the string embedding map, its induced worldsheet metric is ${\gamma }_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$. The Nambu–Goto action isThe first variation uses $\delta {\gamma }_{ab}=2{\partial }_{(a}{X}_{\mu }{\partial }_{b)}\delta X^{\mu }$ and therefore givesAn integration by parts yields the bulk Nambu–Goto equations of motion and the endpoint term:For a closed string, periodicity cancels the endpoint term. For a freely moving open string, arbitrary endpoint variations require zero momentum flux, which is the free-end string boundary condition.

Choose orthogonal conformal coordinates on the nondegenerate part of the worldsheet, so that ${\gamma }_{\tau \sigma }=0$ and ${\gamma }_{\tau \tau }+{\gamma }_{\sigma \sigma }=0$. The Nambu–Goto equations of motion then become a wave equation, with the accompanying Virasoro constraints:Here a dot and a prime mean ${\partial }_{\tau }$ and ${\partial }_{\sigma }$. The last two equations are essential: a solution of the wave equation alone need not be a relativistic fundamental string motion.

For a free-ended open string, take $0\le \sigma \le \pi$ and $a>0$, and setwith the other coordinates constant. Every component satisfies the wave equation, andMoreover $X^{\prime }=0$ at both endpoints, giving the Neumann boundary condition. At fixed target time $t=a\tau$, the spatial image is the straight segmentThus its angular velocity is $1/a$. The speed at coordinate $r$ is $|r|/a$, so the endpoints have the null motion of a free string endpoint. The induced worldsheet metric degenerates there; the conserved densities below have finite limits, so the endpoint degeneracy causes no divergent charge.

In these coordinates ${\Pi }^{\tau \mu }=T{\dot{X}}^{\mu }$. The target-space Noether charges of a string give the energy, momentum and angular momentum:Since this is the centre-of-momentum frame, the rest mass is $M=E$. Eliminating $a$ proves the free-ended straight-string Regge relation:A spacetime Lorentz transformation of the solution changes its frame, not this rest-frame relation.

There is also a folded closed string version: use the same formula over $0\le \sigma \le 2\pi$. The spatial segment is then covered twice, with folds at $\sigma =0,\pi$, and the closed-string mode expansion is periodic. Each charge is twice its open string value, soThe folds also have degenerate induced metric and are understood through the finite canonical densities or a limiting smooth motion. The two classical Regge trajectories have different slopes because the closed string has two coincident branches. No quantum string intercept enters this classical calculation.

In a point-particle description of quantum gravity, interactions of a massless spin-two field occur at localized vertices of Feynman diagrams. The gravitational coupling has negative mass dimension for spacetime dimension greater than two, so short-distance loop integrations generate higher-derivative counterterms that cannot all be absorbed into the Einstein action. A string worldsheet instead joins incoming and outgoing strings by a smooth surface. In particular, a closed string splits or joins through a smooth pair-of-pants surface. There is no invariantly distinguished point at which every part of an extended string interacts. The choice of a time slicing can draw a joining point, but it is not a physical pointlike vertex. This removes the point-interaction geometry responsible for arbitrarily localized interactions and introduces the string length ${\ell }_s=\sqrt{{\alpha }^{\prime }}$.

This geometric explanation is supported by the organization of the Polyakov path integral. Introducing a worldsheet metric makes the Nambu–Goto action amenable to gauge fixing. In the Euclidean integral, metrics on a fixed oriented surface are divided by worldsheet diffeomorphisms and Weyl transformations. What remains is a Riemann surface with a conformal structure, together with finitely many worldsheet moduli and the positions of external string vertex operators. The determinants from fixing the metric are represented by Faddeev-Popov ghosts. Schematically, for $n$ closed-string external states,Here ${\mathcal{M}}_{g,n}$ is the moduli space of genus-$g$ punctured surfaces. The dilaton weights a surface by its Euler characteristic $\chi =2-2g$, giving $g_s^{-\chi }$ before the normalization of external states. Every additional handle multiplies the contribution by $g_s^2$. The string genus expansion therefore plays the role of the ordinary loop expansion, with the sphere giving tree level and the torus giving one loop. The critical dimension cancels the worldsheet Weyl anomaly, making this reduction to conformal geometry consistent.

The massless symmetric traceless closed-string state is a graviton, and its long-wavelength interactions reproduce gravity. At distances comparable with ${\ell }_s$, however, an infinite tower of excited states contributes. This is more than a single particle form factor: the same amplitude contains graviton exchange, massive higher-spin exchange and string-scale softness.

The dimensionless variables in the given Virasoro–Shapiro amplitude correspond to $s={\alpha }^{\prime }S/4$, $t={\alpha }^{\prime }T/4$, $u={\alpha }^{\prime }U/4$, where capital letters denote the usual dimensionful Mandelstam invariants. Its constraint $s+t+u=-4$ identifies the external particles as the closed bosonic tachyons, not four gravitons. Nevertheless its intermediate massless pole directly exhibits graviton exchange. The overall normalization will be suppressed below; no additional copy of the amplitude formula is needed.

First, the amplitude is symmetric under permutations of $s,t,u$, expressing crossing symmetry. The poles in any channel areFor generic $t$, the residue can be calculated with the Gamma function pole at $-n$ and its recurrence:The product is $1$ when $n=0$. Indeed, the pole factor contributes $(-1{)}^{n+1}/n!$, while the two remaining Gamma ratios contribute $(-1{)}^n(t+2{)}_n^2/n!$. A residue polynomial of degree $2n$ shows exchange of spins up to $2n$, with leading Regge trajectory $J=2+2s$. At $n=1$, the massless residue is $-(t+2{)}^2$, containing the spin-two exchange. By crossing, the massless $t$-channel pole isfor generic $s$. Thus at large $s$ the pole behaves as $s^2/t$, the characteristic energy dependence of graviton exchange. Its massless propagator produces the long-range gravitational interaction. At low energies the massive poles can instead be expanded into local higher-derivative corrections to the effective gravitational action.

At the string scale one must retain the whole tower. The Gamma reflection formula rewrites the amplitude exactly asAt large $s$ and fixed $t$, away from the resonance poles or with a specified complex continuation, the Gamma ratio scales as $s^{t+1}$. Hence the Regge limit scales as $s^{2+2t}$, up to the signature factor and the displayed $t$-dependent coefficient. In particular, a fixed nonzero momentum transfer changes the point-graviton power law by a factor $s^{2t}$: the graviton belongs to an entire trajectory rather than remaining an isolated spin-two exchange at arbitrarily high energy.

There is stronger suppression when the scattering angle is fixed. Put $t\simeq -as$, $u\simeq -(1-a)s$ with $0<a<1$. Applying Stirling formula to the Gamma functions, after the same treatment of real-axis poles, gives the leading smooth envelopeThe leading $s\log s$ terms cancel and the remaining entropy-like combination is negative. In physical units the exponent is $-{\alpha }^{\prime }Sf(a)/2$. This fixed-angle softness of the closed-string amplitude is absent for an elementary pointlike gravitational vertex. Equivalently, short-distance gravitational scattering probes additional string degrees of freedom. In a four-dimensional long-distance effective setting, a massive exchanged mode contributes a Yukawa-type correction proportional to $e^{-M_{n}r}/r$, with couplings and tensor factors depending on the external states. Such terms are suppressed for $r\gg {\ell }_s$ and become relevant for $r\sim {\ell }_s$. The amplitude establishes the spectrum and short-distance change; it does not by itself give a universal static potential for arbitrary sources. Its bosonic tachyon also signals an unstable vacuum, so it must not be treated as a complete stable model of a gravitational force.

Finally, the modular group prevents overcounting equivalent surfaces and reorganizes the loop ultraviolet region. A torus is $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$ with $\mathrm{Im}\tau >0$. A change of lattice basis identifiesThus the one-loop integral uses a fundamental domain for $\mathrm{PSL}(2,\mathbb{Z})$, for exampleThe measure $d^2\tau /(\mathrm{Im}\tau {)}^2$ is invariant, and the physical integrand, including matter, ghosts and external insertions, must respect torus modular invariance. In this domain $\mathrm{Im}\tau \ge \sqrt{3}/2$. The region of arbitrarily small proper time that causes ultraviolet divergences in a particle loop is therefore not an independent part of the string integral; a modular transformation relates it to another description already represented in the domain.

This does not remove every possible divergence. The cusp $\mathrm{Im}\tau \to \infty$ is a long tube, and factorization through it can produce infrared divergences, massless tadpoles or the bosonic tachyon divergence. At higher genus, degenerating cycles similarly factorize into propagating intermediate string states. The essential improvement is the relation between worldsheet geometry, the full spectrum and the quotient by large diffeomorphisms; infrared and unstable-background problems remain separate questions. A short-distance expansion that keeps only the graviton discards precisely this structure.

Let $q=\sigma /L$, with $L>0$, and let $Z=X^1+i{X}^2$. Direct differentiation gives $\ddot{Z}=Z^{\prime \prime }=-Z/L^2$, while the time coordinate also satisfies the wave equation. Using the Minkowski metric,Thus both Virasoro constraints hold, and the induced worldsheet metric is $g_{\mu \nu }={\cos }^{2}q\mathrm{diag}(-1,1)$. The configuration solves the Nambu–Goto action equations in its interior.

Place the fixed end at $\sigma =0$. The first free endpoint has $Z^{\prime }=e^{it/L}\cos q=0$ at $q=\pi /2$. For the usual single unfolded rotating Nambu–Goto string,This parameter interval need not equal the earlier interval $[0,\pi ]$, since that interval was a coordinate convention. The proper length is measured on a constant-$X^0$ slice; the local motion is perpendicular to the string, so there is no longitudinal Lorentz contraction.

All points have the same phase $t/L$, hence the segment rotates rigidly with angular velocity $\Omega =1/L$. At radius $r=L\sin q$, its speed is $v=r/L$. Therefore the free tip moves at the speed of light. This is consistent with its Neumann boundary condition: $X^{\prime }=0$ and the Virasoro constraints imply ${\dot{X}}^2=0$ at the free tip. The null endpoint is a boundary limit, not a nondegenerate interior point.

In this conformal gauge, $P_m=T{\dot{X}}_m$, so the energy density per $\sigma$ is $-P_0=T$. Equivalently, the energy density per proper length is $T/\sqrt{1-r^2/L^2}$. Thusand the rotational kinetic energy isThe angular momentum about the fixed point isTherefore the Regge trajectory isThe endpoint conditions alone also admit folded extensions with parameter length $(n+\frac{1}{2})\pi L$, $n\ge 0$. Their proper length, energy and angular momentum are $(2n+1)$ times the values above, and $\beta =1/[(2n+1)\pi T]$. The displayed answer uses the implicit unfolded-segment convention.

At closed-string level $N=\tilde{N}$, the bosonic string mass spectrum isThe state on the leading Regge trajectory uses only level-one oscillators and has maximal spin $J=N+\tilde{N}=2N$. Eliminating $N$ givesBecause $N$ is unbounded, the spectrum contains infinitely many particles of increasing mass and spin. The intercept $J=2$ at $M^2=0$ includes the graviton.

A rigidly rotating stretched solution iswith all other coordinates constant. Each coordinate obeys the wave equation and $X^{\prime }=0$ at $\sigma =0,\pi$. Directly, $\dot{X}\cdot X^{\prime }=0$ andso both Virasoro constraints hold. Its energy and planar angular momentum areConsequently $J=M^2/(2\pi T)={\alpha }^{\prime }{M}^2$, the classical leading open-string Regge trajectory.

A straight rigidly rotating classical string has speed of light at a free endpoint. Its energy and angular momentum satisfy a Regge trajectory. A segment with one anchored and one free end has $J=E^2/(\pi T)$, twice the slope of the string with two free ends.