| Summary: | In this paper, we investigate the four classical tests of general relativity in the non-commutative (NC) gauge theory of gravity. Using the Seiberg–Witten (SW) map and the star product, we calculate the deformed metric components <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mover accent="true"><mi>g</mi><mo>^</mo></mover></mrow><mi>μν</mi></msub><mo>(</mo><mi>r</mi><mo>,</mo><mi>Θ</mi><mo>)</mo></mrow></semantics></math></inline-formula> of the Schwarzschild black hole (SBH). The use of this deformed metric enables us to calculate the gravitational periastron advance of mercury, the red shift, the deflection of light, and time delays in the NC spacetime. Our results for the NC prediction of the gravitational deflection of light and time delays show a newer behavior than the classical one. As an application, we use a typical primordial black hole to estimate the NC parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>Θ</mi></semantics></math></inline-formula>, where our results show <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Θ</mi><mi mathvariant="italic">phy</mi></msup><mo>≈</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>34</mn></mrow></msup><mo> </mo><mi mathvariant="normal">m</mi></mrow></semantics></math></inline-formula> for the gravitational red shift, the deflection of light, and time delays at the final stage of inflation, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Θ</mi><mi mathvariant="italic">phy</mi></msup><mo>≈</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>31</mn></mrow></msup><mo> </mo><mi mathvariant="normal">m</mi></mrow></semantics></math></inline-formula> for the gravitational periastron advance of some planets from our solar system.
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