Testing Born–Infeld f(T) teleparallel gravity through Sgr $$\hbox {A}^\star $$ observations

Abstract We use observational data from the S2 star orbiting around the Galactic Center to constrain a black hole solution of extended teleparallel gravity models. Subsequently, we construct the shadow images of Sgr $$\hbox {A}^{\star }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>A</mml:mtext> <mml:mo>⋆</mml:mo> </mml:msup> </mml:math> black hole. In particular, we constrain the parameter $$\alpha =1/\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>λ</mml:mi> </mml:mrow> </mml:math> which appears in the Born–Infeld f ( T ) model. In the strong gravity regime we find that the shadow radius increases with the increase of the parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> . Specifically, from the S2 star observations, we find within $$1\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> that the parameter must lie between $$0 \le \alpha /M^2 \le 6 \times 10^{-4}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>α</mml:mi> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>≤</mml:mo> <mml:mn>6</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . Consequently, we used the best fit parameters to model the shadow images of Sgr $$\hbox {A}^{\star }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>A</mml:mtext> <mml:mo>⋆</mml:mo> </mml:msup> </mml:math> black hole and then using the Gauss-Bonnet theorem we analysed the deflection angle for leading order expansions of the parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> . It is found that within the parameter range, these observables are very close to the Schwarzschild case. Furthermore, using the best fit parameters for the Born–Infeld f ( T ) model we show that the angular diameter is consistent with recent observations for the Sgr $$\hbox {A}^{\star }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>A</mml:mtext> <mml:mo>⋆</mml:mo> </mml:msup> </mml:math> with angular diameter $$(51.8 \pm 2.3) \mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>51.8</mml:mn> <mml:mo>±</mml:mo> <mml:mn>2.3</mml:mn> <mml:mo>)</mml:mo> <mml:mi>μ</mml:mi> </mml:mrow> </mml:math> arcsec and difficult to be distinguished from the GR. For the deflection angle of light, in leading order terms, we find that the deflection angle expressed in the ADM mass coincides with the GR, but the ADM mass in the Born–Infeld f ( T ) gravity increases with the increase of $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and the overall deflection angle is expected to me greater in f ( T ) gravity. As a consequence of this fact, we have shown that the electromagnetic intensity observed in shadow images is smaller compared to GR.

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