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Cosmological constant corrections to the photon sphere and black hole shadow radii

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Abstract

We review the equations determining the photon sphere radius and the black hole shadow radius, and calculate the cosmological constant corrections arising when the dark energy action has the usual form, and when dark energy arises from a Weyl scaling invariant dark energy action. For black hole targets of the Event Horizon Telescope, the corrections are very small.

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Notes

  1. The photon sphere equation Eq. (4) can also give spacelike hypersurfaces (for example, in the case of the Reissner-Nordström metric [5]), but these are excluded by the specification of a timelike hypersurface in the general definition of a photon surface.

  2. The first and second lines of Table 1 have a different status with respect to a series expansion in \(\Lambda \). The first line comes from the conventional dark mater action of Eq. (11), which leads to the Schwarzschild-de Sitter metric and is exact in \(\Lambda \). So the result \(r_{ph}=3M\) obtained from the first line is exact to all orders in \(\Lambda \). The second line comes from the order \(\Lambda \) term in the power series expansion that is calculated in [9]. Terms of order \(\Lambda ^2\) and higher were not calculated in [9] and are absent from the second line of Table 1, so it is only consistent to calculate the \(D_B\) correction to \(r_{ph}=3M\) to order \(\Lambda \), dropping higher order terms. That is what is done in Eq. (15).

  3. Rewriting Eq. (7) as \((d/dr)(B(r)/r^2)=0\), which yields Eq. (8) when expanded out, we see by inspection that since \(B(r)/r^2=1/r^2-2M/r^3-C_B \Lambda +D_B \Lambda M /r\), the constant \(C_B\) term vanishes when the first derivative is taken.

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Authors and Affiliations

  1. Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ, 08540, USA

    Stephen L. Adler

  2. Mathematics Department, Drexel University, 33rd and Market Streets, Philadelphia, PA, 19104, USA

    K. S. Virbhadra

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  1. Stephen L. Adler
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  2. K. S. Virbhadra
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Correspondence to Stephen L. Adler.

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Adler, S.L., Virbhadra, K.S. Cosmological constant corrections to the photon sphere and black hole shadow radii. Gen Relativ Gravit 54, 93 (2022). https://doi.org/10.1007/s10714-022-02976-7

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  • DOI: https://doi.org/10.1007/s10714-022-02976-7

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