The relativistic blackbody spectrum [1] suggests the intriguing possibility that a luminous astrophysical body can be rendered optically invisible to the human eye by relativistic Doppler shifting the wavelengths of maximum intensity from the visible frequency range to above or below the frequency thresholds of human vision.1 In honor of the 100^th anniversary of Einstein’s general relativity, this note examines, as a gedanken experiment, the specific conditions under which this effect would occur.2

Furthermore, relativistic blackbody radiators will emit spectral radiances which are increased (in the case of approaching) or decreased (in the case of receding), due to temperature inflation and relativistic beaming. By considering in the gedanken experiment the relativistic blackbody spectrum, the proper distances can be determined at which the apparent magnitude of a blackbody radiator is greater (i.e. dimmer) than approximately 6.5 (the threshold of vision for the typical unaided human eye).

Additionally, laboratory tests of the sensitivity of the unassisted human eye are described, and this paper asserts that the Judd & Voss CIE 1978 photopic luminous efficiency function would not be applicable to the situation of LABs due to the much greater luminosity than in the laboratory tests.

The relationship between absolute magnitude, apparent magnitude, and distance to an arbitrary stationary blackbody radiation source has been well established and is given by:

where M is the absolute magnitude of any blackbody radiator, m is its apparent magnitude, and z is the distance to the observer in parsecs. Also, in terms of luminosity,

where M_o is the absolute magnitude of a reference star (e.g. the sun), L is the luminosity of the radiation source at an arbitrary distance z, and L_o is the absolute luminosity of that source3.

Equating Equations (1) and (2) yields:

Thus, for the sun, M_o = 4.83, , and for it to be invisible to the naked eye in the [nearly] total blackness of interstellar space, m = 6.5 [2] (discussed in Section 4). Therefore, the sun is visible to the unaided eye at distances up to 21.58 pc (70.39 LY).

Sufficiently high speed relativistic motion of blackbody radiators would clearly Doppler shift the wavelengths of maximum luminosity to beyond the human visual range. Therefore, the lower luminosity wavelengths are Doppler shifted into the visible range, and the overall visible luminosity is reduced.

However, in the case of an approaching blackbody, the radiation is relativistically beamed, and the blackbody temperature is “inflated”. Both of these effects serve to increase the luminosity. For a receding blackbody, relativistic beaming (“expanding”) and temperature “deflation” will have the reverse effect. Therefore, Equation (3) becomes:

where and L_o are the luminosities in the relativistic and rest frames respectively4.

Luminosity is obtained by integrating the spectral radiance over frequency and solid angle:

where ν₁ and ν₂ are the mean lower and upper frequencies of ocular visibility. and B_o are the spectral radiances in the relativistic and rest frames respectively, which must be integrated over the appropriate solid angle Ω. The relativistic spectral radiance in frequency space, accounting for Doppler shifting, relativistic beaming, and temperature inflation, was determined by Lee and Cleaver [1] , and is given by5:

where [3] ,

and is the proper absolute temperature, is the relative 4-velocity between the radiation and the observer, is the van Kampen-Israel inverse tem-

perature 4-vector, θ is the angle between and, (fraction of light speed).

The integration of the spectral radiance over all frequencies is straightforward because, with the limits of 0 and ∞, the result is simply π⁴/15. However, the in-band luminosity requires integration over a finite frequency range. Here, the method of Widger and Woodall is followed [4] .

Letting:

and

Also, letting, and from Equations (7) and (8), Equation (9) becomes:

Expanding Equation (12) as a difference of integrals:

Evaluating Equation (13), and re-substituting:

Expanding the solid angle integration, combining sums, and making use of

from Equation (5), the relativistic luminosity in frequency space

(Equation (14)) becomes6:

In the case of approaching the LAB [approximately] directly, a simplification of Equation (15), which cannot be resolved as a closed form function, can be made. Since, θ is very small and. This removes the angular dependence from Equation (11), which reduces to:

Frequently, when evaluating the dΩ integration, the solid angle over which the integration is performed is the solid angle through which the blackbody radiates. However, that is not the case here. The solid angle is that which is subtended by the blackbody

from the vantage point of the observer. Therefore, when, (z is the observer proper distance, and D is the diameter of the blackbody). For a blackbody with a circular x-y cross-section,. Therefore,

Thus,

Similarly,

Combining Equation (3), in terms of relativistic luminosity, with Equations (18) and (19) yields:

Evaluation of the infinite sums is greatly simplified due, in large part, to the rapid convergence of the series as a result of the terms. The smallest value of R (requiring the largest number of summation terms) occurs when V = 0, and from Equa-

tion (11), is. The smallest useful value of would result from an O-

class star, with a surface temperature of ~50,000 K, and at the lowest frequency of human visibility. Table 1 gives the number of summation terms (n) (in Equation (20)) that would be required to produce at least 10 significant figure convergence for.

The visibility to the naked eye of astronomical objects has been discussed extensively in the literature [5] - [9] . The “sky” of interstellar space is considered to be absolutely black, and a viewing port is taken to be, at optical wavelengths, a perfectly transparent aperture that subtends a solid angle of at least the human field of vision and with a magnification of 1.

The efficiency by which photons are used by the retina was accounted for by correcting for the Stiles-Crawford effect of the first (SCE I) and second (SCE II) kind7, photon absorption by the optical media, photopigment absorption of photons, and the photon isomerization efficiency of the photopigment.

For a 22' (diameter), 10 ms, 507 nm monochromatic source, in which, of the ~100 quanta incident upon the retina, 10 to 15 were absorbed by the ~1600 illuminated rods [10] . From this experiment, Packer and Williams determined the rod actinometric, radiometric, and photometric8 absolute thresholds to be 0.35 γ/s, 4.35 × 10⁻⁶ W・m⁻²・sr⁻¹, and 1.33 × 10⁻³ cd・m⁻² respectively [11] . However, also examined was the case of a stimulus which exceeded the visual system’s spatial summation area and temporal integration time, in which the rod actinometric, radiometric, and photometric absolute thresholds are 2.00 × 10⁻⁴ γ/s, 2.47 × 10⁻⁹ W・m⁻²・sr⁻¹, and 7.5 × 10⁻⁷ cd・m⁻² respectively.

However, even accounting for the standard observer’s spectral sensitivity by applying the Judd & Voss CIE 1978 photopic luminous efficiency function, these results are difficult to apply to the scenario presented here because of the enormous disparity between the spectral irradiances of the Hallett test sources [10] and stars.

The frequency range of human vision is slightly variable. However, 4.17 × 10¹⁴ Hz and 7.89 × 10¹⁴ Hz, which correspond to wavelengths of 720 nm and 380 nm respectively, are acceptable approximations of the limits of human vision, and are in keeping with the wavelengths of 700 nm and 390 nm published by Starr [12] . The limiting magnitude of the unassisted human eye is taken to be 6.5 [2] . This figure applies to all visible wavelengths and accounts for eye sensitivity. Consequently, inclusion of the Judd & Voss CIE 1978 photopic luminous efficiency function would not be appropriate.

By making use in this gedanken experiment of the relativistic blackbody spectrum, the velocity profile for the apparent magnitude of a LAB has been determined. Optical invisibility to the unaided eye arises due to the Doppler shifting of the wavelengths of maximum radiance to beyond the limits of human visual sensitivity. Temperature inflation and relativistic beaming can either increase this incident radiance (for an approaching source) or decrease it (for a receding source). By considering the wavelength limits of human vision to be 380 nm and 720 nm, and the limiting magnitude of the unaided human eye to be 6.5, the proper distance versus velocity function for ocular invisibility of relativistic luminous astrophysical bodies has been determined; this profile was determined for the sun.

Whether the physical situations could exist for this effect to be realized is uncertain. As an example, relativistic speeds might be obtainable in the expulsion of a low-mass star from the region of the galactic center as a consequence of a fly-by with the central massive blackhole. Nevertheless, such a star might already appear invisible from earth because of its distance, rather than as a result the relativistic Doppler Effect. The possible realization of this gedanken experiment is an open question.

Lee, J.S. and Cleaver, G.B. (2016) Black Sun: Ocular Invisibility of Relativistic Luminous Astrophysical Bodies. Journal of High Energy Physics, Gravitation and Cosmology, 2, 562-570. http://dx.doi.org/10.4236/jhepgc.2016.24048