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Off-Shell Linear-Dilaton Frame-Density Check

Physical scope

A local off-shell algebraic check of the NS–NS action density in two frames for a smooth linear dilaton on flat string-frame spacetime. It is not a solution, a background-existence claim, an integrated-action claim, or empirical evidence.

Statement

On $U = R^{1, 9}$ with coordinates $x^{M}$ of length dimension, let $G_{s} = η = diag (- 1, 1, \dots, 1)$ , let $B_{2} = 0$ , and let $Φ (x) = λ x^{1}$ for a fixed nonzero constant $λ$ of dimension length $^{- 1}$ . These smooth fields are an off-shell configuration of the adopted local NS–NS model. The string-frame and Einstein-frame Lagrangian densities, each including the common factor $(2 κ_{0}^{2})^{- 1}$ and expressed relative to $d^{10} x$ , agree pointwise and equal $\frac{1}{2 κ_{0}^{2}} 4 λ^{2} e^{- 2 Φ} .$ The configuration is not stationary because its string-frame dilaton Euler coefficient is $D = - 4 λ^{2} \neq 0$ . No integrated action is asserted.

Facts & Assumptions

Given: The flat string-frame metric, linear dilaton, vanishing two-form, and nonzero constant $λ$ of the Statement.

[F1]

Coordinates have length dimension, metric components and $Φ$ are dimensionless, and the curvature and wave-operator conventions are fixed as in the convention item (Supergravity Conventions And Normalizations).

[A1]

The adopted string-frame density is $(2 κ_{0}^{2})^{- 1} \sqrt{- G_{s}} e^{- 2 Φ} (R_{s} + 4 ∣ d Φ ∣_{G_{s}}^{2} - \frac{1}{12} H_{s}^{2})$ (Ten Dimensional NS NS Supergravity Action).

[F2]

For $G_{E} = e^{- Φ / 2} G_{s}$ in ten dimensions, the Einstein-frame density inside $(2 κ_{0}^{2})^{- 1} \sqrt{- G_{E}}$ is $R_{E} - \frac{1}{2} ∣ d Φ ∣_{G_{E}}^{2} - \frac{1}{12} e^{- Φ} H_{E}^{2} - \frac{9}{2} □_{E} Φ$ (String Einstein Frame NS Action Calculation).

[F3]

The scalar-curvature transformation under $\tilde{g} = e^{2 σ} g$ is the general $n$ -dimensional identity of Scalar Curvature Under Conformal Rescaling.

[F4]

A stationary configuration of the adopted action satisfies $D = R + 4 □ Φ - 4 ∣ d Φ ∣^{2} - \frac{1}{12} H^{2} = 0$ (Ten Dimensional NS NS Field Equations).

Proof

technique · direct; evaluate each invariant and density in the string frame, then use the conformal formula and the Einstein-frame action for the same fields

1.1givenF1A1algebra

In the string frame $R_{s} = 0$ , $H = 0$ , $∣ d Φ ∣_{G_{s}}^{2} = λ^{2}$ , and $\sqrt{- G_{s}} = 1$ , because $η$ is constant, $G_{s}^{11} = 1$ , and $\partial_{1} Φ = λ$ is the only nonzero derivative. Hence [A1] gives the string-frame density $(2 κ_{0}^{2})^{- 1} 4 λ^{2} e^{- 2 Φ}$ . Its dimension is length $^{- 10}$ : $κ_{0}^{2}$ has dimension length $^{8}$ and $λ^{2}$ has dimension length $^{- 2}$ , as required for a density relative to $d^{10} x$ .

2.1F1F3step 1.1algebra

Set $G_{E} = e^{- Φ / 2} η = e^{2 σ} η$ with $σ = - Φ / 4$ . Since $R_{η} = 0$ , $□_{η} σ = 0$ , and $∣ d σ ∣_{η}^{2} = λ^{2} / 16$ , [F3] at $n = 10$ gives $R_{E} = e^{Φ / 2} [- 72 λ^{2} / 16] = - \frac{9}{2} λ^{2} e^{Φ / 2}$ . The inverse metric and determinant scale as $G_{E}^{11} = e^{Φ / 2}$ and $\sqrt{- G_{E}} = e^{- 5 Φ / 2}$ .

3.1step 2.1F2algebra

Therefore $∣ d Φ ∣_{G_{E}}^{2} = λ^{2} e^{Φ / 2}$ . By [F2], $□_{E} Φ = (\sqrt{- G_{E}})^{- 1} \partial_{M} (\sqrt{- G_{E}} G_{E}^{M N} \partial_{N} Φ)$ . Only $M = N = 1$ contributes; $\sqrt{- G_{E}} G_{E}^{11} \partial_{1} Φ = e^{- 2 Φ} λ$ , whose $x^{1}$ -derivative is $- 2 λ^{2} e^{- 2 Φ}$ . Dividing by $e^{- 5 Φ / 2}$ gives $□_{E} Φ = - 2 λ^{2} e^{Φ / 2}$ .

4.1F2step 1.1step 2.1step 3.1algebra

With $H = 0$ , substituting the three quantities into [F2] gives $R_{E} - \frac{1}{2} ∣ d Φ ∣_{G_{E}}^{2} - \frac{9}{2} □_{E} Φ = (- \frac{9}{2} - \frac{1}{2} + 9) λ^{2} e^{Φ / 2} = 4 λ^{2} e^{Φ / 2}$ . Multiplying by $\sqrt{- G_{E}} = e^{- 5 Φ / 2}$ and $(2 κ_{0}^{2})^{- 1}$ gives the Einstein-frame density $(2 κ_{0}^{2})^{- 1} 4 λ^{2} e^{- 2 Φ}$ , exactly the string-frame value in step 1.1. The divergence term is included in this comparison.

5.1givenF1F4step 1.1algebra∎

In the string frame $R_{s} = 0$ , $□_{s} Φ = 0$ , $H = 0$ , and $X_{s} = λ^{2}$ , so $D = 0 + 0 - 4 λ^{2} - 0 = - 4 λ^{2} \neq 0$ . By [F4], the configuration fails the dilaton stationarity equation and is therefore off shell, as claimed.

Off-Shell Linear-Dilaton Frame-Density Check

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources