Question re: time dilation equations

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SUMMARY

This discussion focuses on the differences between two sets of time dilation equations: Set A, which uses \(\Delta t'\) for time observed in a moving frame, and Set B, which uses \(\Delta t_0\) for proper time. The equations discussed are \(\Delta t' = \frac{\Delta t}{\sqrt{1-(v/c)^2}}\) and \(\Delta t' = \frac{\Delta t}{\sqrt{1-\frac{2GM}{rc^2}}}\) for kinematic and gravitational time dilation, respectively. The inconsistency in symbol usage highlights the lack of standardization in physics notation, necessitating careful reading of context to understand definitions. Additionally, the discussion includes a request for numerical examples of gravitational time dilation as experienced by an observer on Earth's surface.

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For both time dilation equations (kinematic and gravitational) I have often seen

\Delta t^\prime = \frac{\Delta t}{\sqrt{1-(v/c)^2}}

and

\Delta t' = \frac{\Delta t}{\sqrt{1-\frac{2GM}{rc^2}}}

I'll calls these equations as "Set A"

------------------------

And at other times as

\Delta t = \frac{\Delta t_0}{\sqrt{1-(v/c)^2}}

and

\Delta t = \frac{\Delta t_0}{\sqrt{1-\frac{2GM}{rc^2}}}

I'll calls these equations as "Set B"

---

I'm no physics (or mathematics) major (just a dilettante; that's why I'm asking.) Why use t' (on the left side of the equals sign for "Set A" equations) but t0 (on the right side of the equals sign for "Set B" equations)? I realize an equation is only good as far as you can interpret it. And so I know these are the same equations, but why the difference?

Thanks. :)
 
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It would be nice if all physicists agreed on the symbols used to represent physical quantities in equations. Unfortunately they don't. In case of apparent conflict, you need to read the text surrounding the equations carefully, to find out how each author defines his symbols.
 
jtbell said:
It would be nice if all physicists agreed on the symbols used to represent physical quantities in equations. Unfortunately they don't. In case of apparent conflict, you need to read the text surrounding the equations carefully, to find out how each author defines his symbols.
Of course. Universality of equation symbology would be nice, but that'll never happen. :-)

---

Completely tangential to my OP: I can easily plug numbers into the kinematic time dilation equation without even thinking about it. But I've never really involved myself with the gravitation time dilation equation until recently. I'm still trying to get a firm grasp on it. With that said, can anyone provide me an example of the numbers to use in the gravitational time dilation equation regarding someone on the surface of the Earth as viewed by an observer very far away from earth?

Thanks, your response(s) are much appreciated.
 

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