Why Is the tt Component of the Metric Interpreted Differently in This Paper?

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The discussion centers on the interpretation of the tt component of the metric in a specific paper, where the authors define g_{tt} as g(r) rather than the expected f(r)g(r). This discrepancy raises questions about the authors' reasoning and the implications for the overall metric. Participants seek clarification on the authors' methodology and the context of the paper. Understanding this interpretation is crucial for accurately applying the metric in relevant calculations. The conversation highlights the importance of clear definitions in theoretical physics.
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I am reading a paper and the authors read the tt component of the metric from the line element ds^2=f(r)[g(r) dt^2+h(r) dr^2] as g_{tt}=g(r) instead of (what I expect to be) g_{tt}=f(r) g(r)

Could somebody please explain to me why? Thank you.
 
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This seems strange. What is the paper?
 

Thread 'Am I understanding the concept of proper frame of reference?'

MOVING CLOCKS In this section, we show that clocks moving at high speeds run slowly. We construct a clock, called a light clock, using a stick of proper lenght ##L_0##, and two mirrors. The two mirrors face each other, and a pulse of light bounces back and forth betweem them. Each time the light pulse strikes one of the mirrors, say the lower mirror, the clock is said to tick. Between successive ticks the light pulse travels a distance ##2L_0## in the proper reference of frame of the clock...
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