Radial inverse metric components do not identify null surfaces

We investigate the conditions under which a hypersurface becomes null through the use of coordinate transformations. We demonstrate that, in static spacetimes, the correct criterion for a surface to be null is <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi>g</a:mi></a:mrow><a:mrow><a:mi>t</a:mi><a:mi>t</a:mi></a:mrow></a:msub><a:mo>=</a:mo><a:mn>0</a:mn></a:mrow></a:math>, rather than <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>g</c:mi><c:mrow><c:mi>r</c:mi><c:mi>r</c:mi></c:mrow></c:msup><c:mo>=</c:mo><c:mn>0</c:mn></c:math>, in agreement with the results of Vollick. We further show that, if a Kruskal-like coordinate exists, the proxy condition <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>g</e:mi><e:mrow><e:mi>r</e:mi><e:mi>r</e:mi></e:mrow></e:msup><e:mo>=</e:mo><e:mn>0</e:mn></e:math> is equivalent to <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>g</g:mi><g:mrow><g:mi>t</g:mi><g:mi>t</g:mi></g:mrow></g:msub><g:mo>=</g:mo><g:mn>0</g:mn></g:math> if <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mo>∂</i:mo><i:mi>r</i:mi></i:msub><i:msub><i:mi>g</i:mi><i:mrow><i:mi>t</i:mi><i:mi>t</i:mi></i:mrow></i:msub><i:mo>≠</i:mo><i:mn>0</i:mn></i:math> and both <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>g</k:mi><k:mrow><k:mi>r</k:mi><k:mi>r</k:mi></k:mrow></k:msup></k:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>g</m:mi><m:mrow><m:mi>t</m:mi><m:mi>t</m:mi></m:mrow></m:msub></m:math> vanish at the same rate near the horizon. Our method extends naturally to axisymmetric stationary spacetimes, for which we demonstrate that the condition <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mrow><o:mi>det</o:mi><o:mo stretchy="false">(</o:mo><o:msub><o:mrow><o:mi>h</o:mi></o:mrow><o:mrow><o:mi>a</o:mi><o:mi>b</o:mi></o:mrow></o:msub><o:mo stretchy="false">)</o:mo><o:mo>=</o:mo><o:mn>0</o:mn></o:mrow></o:math> for the induced metric on a null hypersurface is recovered. By contrast with the induced metric approach, our method provides a physical perspective that connects the general null condition with its underlying relationship to photon geodesics.