The Blackbody Radiation Spectrum Follows from Zero-Point Radiation and the Structure of Relativistic Spacetime in Classical Physics

Abstract

The analysis of this article is entirely within classical physics. Any attempt to describe nature within classical physics requires the presence of Lorentz-invariant classical electromagnetic zero-point radiation so as to account for the Casimir forces between parallel conducting plates at low temperatures. Furthermore, conformal symmetry carries solutions of Maxwell’s equations into solutions. In an inertial frame, conformal symmetry leaves zero-point radiation invariant and does not connect it to non-zero-temperature; time-dilating conformal transformations carry the Lorentz-invariant zero-point radiation spectrum into zero-point radiation and carry the thermal radiation spectrum at non-zero temperature into thermal radiation at a different non-zero temperature. However, in a non-inertial frame, a time-dilating conformal transformation carries classical zero-point radiation into thermal radiation at a finite non-zero-temperature. By taking the no-acceleration limit, one can obtain the Planck radiation spectrum for blackbody radiation in an inertial frame from the thermal radiation spectrum in an accelerating frame. Here this connection between zero-point radiation and thermal radiation is illustrated for a scalar radiation field in a Rindler frame undergoing relativistic uniform proper acceleration through flat spacetime in two spacetime dimensions. The analysis indicates that the Planck radiation spectrum for thermal radiation follows from zero-point radiation and the structure of relativistic spacetime in classical physics.

Appendix: Comment on the Conformal Transformations in a Rindler Frame in Two Spacetime Dimensions

An anonymous referee for this manuscript has remarked that although “the observation of two types of conformal transformation naturally associated with a Rindler frame is novel and interesting; … the treatment of those transformations in Sect. 2 is opaque and does not bring out their geometrical meaning.” On this account,a paraphrase of the referee’s discussion has been added here.

The statement in the paragraph following (19) that “the coordinate transformation from an inertial frame to a Rindler frame is not conformal” is technically correct for (2), (3), (4), (9) as written, but it is misleading in that if one makes the further one-dimensional coordinate change ξ=e ^u, the result is conformal (in dimension 2);

$$ds^{2}=e^{2u}\bigl(d\eta^{2}-du^{2}\bigr)$$

(60)

and the (massless 2D) wave equation becomes ($\partial_{u}^{2}-\partial_{\eta }^{2})\phi=0$ (times a conformal factor). In these coordinates, the “space-dilating” transformation (20) becomes

$$\eta\rightarrow\eta,\qquad u\rightarrow u+\ln\sigma $$

(61)

and the “time-dilating” transformation (24) becomes

$$\eta\rightarrow\sigma\eta,\qquad u\rightarrow\sigma u $$

(62)

Transformation (20) is completely equivalent, via (2)–(3), to the original uniform inertial dilation mentioned above (16) (restricted to the Rindler wedge, of course). This is the precise counterpart of the fact that a uniform dilation of the Euclidean plane amounts in polar coordinates to a rescaling of r leaving θ unchanged. (From this point of view, the result of Sect. 4.4 is totally unsurprising.)

An inertial dilation does not need to have its fixed point at the origin; any other point in spacetime could do as well:

$$x-x_{0}\rightarrow\sigma(x-x_{0}),\qquad t-t_{0}\rightarrow \sigma(t-t_{0})$$

(63)

This is the key to the significance of (24) or (62). The fixed point (η,u)=(0,0) (ξ=1) corresponds to (x ⁰,x ¹)=(0,1). Expand (2)–(3) (after and before composition with (62)) to first order in η and u:

(64)

which is (62) again. So (62) is the first-order approximation in Rindler coordinates to the inertial dilation around (x ⁰,x ¹)=(0,1) (or vice versa). Such a transformation could be made around the origin of any nonsingular coordinate system and just constitutes a shrinking of the scale of one’s attention (or of a solution, in the active interpretation) down toward an infinitesimal neighborhood of the point with coordinates (0,0). In that limit, both curvature and (as in this case) artificial inertial forces vanish.

In short, “space-dilating conformal transformation” is inertial dilation about the vertex of the Rindler wedge, and “time-dilating conformal transformation” is the lowest-order local approximation to inertial dilation about a point in the interior.