Time Dilation & Differential Aging: Defined

[exmarine]

I’ve been told that “differential aging” and “time dilation” are two different things. I had thought that one was the integral of the other. Can anyone give a PRECISE (mathematical!) definition of each, what the distinctions are between them? Thanks.

(How the heck am I supposed to know what thread level to assign to my questions??)

 

[stevendaryl]

I’ve been told that “differential aging” and “time dilation” are two different things. I had thought that one was the integral of the other. Can anyone give a PRECISE (mathematical!) definition of each, what the distinctions are between them? Thanks.

(How the heck am I supposed to know what thread level to assign to my questions??)

If Alice and Bob are moving at a relative speed of 99% of the speed of light, then Alice will view Bob's clock to be dilated, while Bob will view Alice's clock to be dilated. It's only if they eventually reunite that the phrase "differential aging" is applicable. Time dilation is a coordinate effect, while differential aging is independent of any coordinate system.

I would not say that differential aging is the integral of time dilation---I would say that aging is. To have differential aging, you need to compare two different ages.

So, for mathematical definitions:

1. Time dilation is a property of a spacetime path relative to a coordinate system: $\mathcal{F} = \frac{d\tau}{dt} =$ the rate at which the proper time $\tau$ for the path increases as a function of coordinate time $t$. For inertial Cartesian coordinates, $\mathcal{F} = \sqrt{1-\frac{v^2}{c^2}}$ where $v$ is the spatial velocity for the path, as measured in those coordinates.
2. Differential aging is a property of a pair of spacetime paths and a pair of spacetime points (or "events"): If you have two spacetime paths $\mathcal{P}_1$ and $\mathcal{P}_2$, and a pair of events $e_{i}$ and $e_{f}$ such that both paths pass through both points, then the differential aging for the paths is $\int_{e_i}^{e_f} d \tau_2 - \int_{e_i}^{e_f} d \tau_1$. This quantity is independent of coordinate systems. But given a coordinate system, it can be written as a single integral: $\int_{t_i}^{t_f} (\mathcal{F}_2 - \mathcal{F}_1) dt$.

 

[Nugatory]

"Differential aging": the elapsed time along different paths in spacetime between the same two events will, in general, be different because the "lengths" of the paths will, in general, be different. The twin paradox is the classic example: the traveling twin takes a shorter path through spacetime between the event "twins separate" and the event "twins reunite" so less time passes for him and he ages less.

"Time dilation" (not to be confused with "gravitational time dilation"): a clock that is not in rest in a given frame will run slow compared with a clock that is at rest in that frame. Because the two clocks are moving relative to one another, we're always working with at least three events: "both clocks are in the same place and both read midnight"; "clock A is somewhere and reads 1:00 AM"; "clock B is somewhere else and reads 1:00 AM". Because the two clocks are not collocated when either reads 1:00 AM we have to consider relativity of simultaneity. The "A reads 1:00 AM" event happens before the "B reads 1:00 AM" event (B's clock is slow) using the the frame in which A is at rest; but it is the other way around and A's clock is the slow one using the frame in which B is at rest.

You'll find a bunch more discussion in the "Similar Discussions" at the bottom of the page.

 

[Nugatory]

(How the heck am I supposed to know what thread level to assign to my questions??)

It's the level of the answer that you want. "B" will get an answer using no math beyond algebra, trig, and Euclidean geometry, maybe single-variable calculus. "I" says that you're OK with multiple-variable and vector calculus, have been through basic college-level physics.

Stevendaryl and I are saying the same thing, but his answer is "I" level and mine was "B" level.

 

[exmarine]

THANKS!