Time Dilation Notation in SR: Clarifying the Differential Form

[connorp]

So I have seen time dilation written as all three of the following: t=τγ, Δt=Δτγ, dt=dτγ. I'm assuming this not to be the case, but just wanted to clarify that the third (differential) notation does not imply that t=∫τγ? That really wouldn't make sense (to me at least), so I'm assuming that that notation is just a convention and has no mathematical significance over the first equation (t=τγ)?

 

[PeterDonis]

All of these notations are incomplete, and I would not advise using them as actual formulas in any calculation. To actually compute correct answers, you need a complete coordinate chart, not just a single coordinate.

 

[ghwellsjr]

So I have seen time dilation written as all three of the following: t=τγ, Δt=Δτγ, dt=dτγ. I'm assuming this not to be the case, but just wanted to clarify that the third (differential) notation does not imply that t=∫τγ? That really wouldn't make sense (to me at least), so I'm assuming that that notation is just a convention and has no mathematical significance over the first equation (t=τγ)?

I take advantage of the second version of that nomenclature in the application I wrote to display in-line scenarios in Special Relativity. Here, for example, is an observer that starts out at rest in the Inertial Reference Frame. The dots mark off 1-nsecs increments of Proper Time. After 4 nsecs of his time, τ, he moves away at 0.6c. Notice how the Coordinate Time deltas for each pair of marks is 1.25 times the Proper Time increment while at the speed of 0.6c. After 5 more nsecs of Proper time, the observer starts traveling at 0.8c with γ=1.667. During this last segment, the Coordinate Time deltas is 1.667 times the Proper Time. I wrote my application so that I could specify each segment of an observer's profile as a specific amount of Proper Time at a specific speed. The application uses Δt=Δτγ along with the specified speed (and Δx=vΔt) to determine where to put the marks on the spacetime diagram:

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The first version of your nomenclature is valid only when the observer starts at the origin and is inertial, at least for the first segment of the scenario.

I think the last version of your nomenclature is also always valid but I don't think the integral is specified correctly.