- #1

Mike_Fontenot

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The CADO equation, that I have described previously, applies to ANY kind of accelerations by the traveling twin. Its only requirement is that the home twin remain inertial.

For the simple limiting cases where the only velocity-changes made by the traveler are INSTANTANEOUS velocity-changes occurring at various instants in the traveler's life, separated by periods of constant velocity by the traveler, the CADO equation becomes especially simple, and especially easy to use.

In that case, it's possible to quickly and easily calculate the instantaneous CHANGE in the age of the home twin, according to the traveler, caused by an instantaneous velocity CHANGE by the traveler.

Denote the instantaneous velocity change of the traveler, at some instant, as "delta(v)". This is just the new velocity v2, minus the old velocity v1. (Velocities are measured in lightyears per year, which are always numbers greater than -1 and less than +1. Negative velocities mean that the twins are moving TOWARD each other.)

The ONLY other thing you need to know is the distance between the twins (according to the home twin), at the instant when the velocity change occurs. Call that distance "L" (measured in lightyears).

The age of the home twin, according to the traveler, is denoted CADO_T. (The acronym "CADO" always refers to the HOME twin's current age, at some given instant of the traveler's life, and the subscripts "_T" and "_H" are added to indicate WHOSE conclusion about the home twin's current age (Traveler or Home twin) we're referring to).

Denote the instantaneous CHANGE in the age of the home twin (according to the traveler), at some given instant of the traveler's instantaneous velocity change, as "delta(CADO_T)". delta(CADO_T) is just the new value of CADO_T, minus the old value of CADO_T ... immediately before and immediately after the instantaneous velocity change.

Then, the simple equation is just

delta(CADO_T) = -L * delta(v).

That's all there is to it.

(In the above equation, I've omitted some factors of c (the speed of light). Since we are using units for which c = 1 lightyear/year, those factors can be ignored ... they are needed only for dimensional correctness).

Note that the magnitude of the instantaneous change in the age of the home twin (according to the traveler), caused by a given magnitude of instantaneous velocity change, is proportional to the separation L of the twins.

In particular, this means that if L = 0 (i.e., if the twins are co-located), the instantaneous velocity change has NO effect on the age of the home twin. That's why the initial and final accelerations by the traveler (in the case of the classic twin "paradox" scenario, where the twins are co-located at the beginning and end of the voyage) don't affect the outcome of the twin "paradox" problem at all ... the classic twin scenario can always be reformulated with no accelerations at the beginning and end of the voyage, without changing the overall outcome.

And because their separation shows up as a proportionality factor in the equation, that means that, for a given velocity change, the effect on the age change of the home twin becomes greater when their separation is larger. In fact, you can see from the equation that the home twin's age can suddenly change by (almost as much as) twice the separation (because delta(v) can have a magnitude as large as 2 minus an infinitesimal amount.).

Here's an example of the use of the equation:

Suppose the twins are 30 lightyears apart at some instant of the traveler's life, and that the traveler's velocity has been constant at v1 = -0.8 for some period of time up until that instant.

The negative sign means that the twins have been moving toward each other. (For simplicity, I'm not bothering to write the units of velocity (lightyears/year)).

Suppose that the traveler then instantaneously changes his velocity to v2 = 0.6. Then

delta(v) = v2 - v1 = (0.6) - (-0.8) = 1.4,

and so we get

delta(CADO_T) = -30 * 1.4 = -42 years.

So, with the given instantaneous velocity change, and with the given separation, the home twin gets YOUNGER by 42 years (according to the traveler), during the traveler's instantaneous change of velocity.

Once you've got the capability to quickly and easily calculate the instantaneous age-changes of the home twin (according to the traveler), caused by instantaneous velocity-changes by the traveler, you can easily get the complete solution for how the home twin's age changes (according to the traveler), during a complete (and fairly complicated) voyage by the traveler, consisting of any arbitrary sequence of instantaneous velocity-changes, separated by periods of constant velocity in between those velocity-changes. The ageing of the home twin, during each of the constant-velocity periods, is easily calculated from the well-known time dilation result, and the home twin's ageing during each of the instantaneous velocity-changes can be easily calculated using the above delta(CADO_T) equation ... the combination of all those amounts of ageing gives the complete solution.

Mike Fontenot

For the simple limiting cases where the only velocity-changes made by the traveler are INSTANTANEOUS velocity-changes occurring at various instants in the traveler's life, separated by periods of constant velocity by the traveler, the CADO equation becomes especially simple, and especially easy to use.

In that case, it's possible to quickly and easily calculate the instantaneous CHANGE in the age of the home twin, according to the traveler, caused by an instantaneous velocity CHANGE by the traveler.

Denote the instantaneous velocity change of the traveler, at some instant, as "delta(v)". This is just the new velocity v2, minus the old velocity v1. (Velocities are measured in lightyears per year, which are always numbers greater than -1 and less than +1. Negative velocities mean that the twins are moving TOWARD each other.)

The ONLY other thing you need to know is the distance between the twins (according to the home twin), at the instant when the velocity change occurs. Call that distance "L" (measured in lightyears).

The age of the home twin, according to the traveler, is denoted CADO_T. (The acronym "CADO" always refers to the HOME twin's current age, at some given instant of the traveler's life, and the subscripts "_T" and "_H" are added to indicate WHOSE conclusion about the home twin's current age (Traveler or Home twin) we're referring to).

Denote the instantaneous CHANGE in the age of the home twin (according to the traveler), at some given instant of the traveler's instantaneous velocity change, as "delta(CADO_T)". delta(CADO_T) is just the new value of CADO_T, minus the old value of CADO_T ... immediately before and immediately after the instantaneous velocity change.

Then, the simple equation is just

delta(CADO_T) = -L * delta(v).

That's all there is to it.

(In the above equation, I've omitted some factors of c (the speed of light). Since we are using units for which c = 1 lightyear/year, those factors can be ignored ... they are needed only for dimensional correctness).

Note that the magnitude of the instantaneous change in the age of the home twin (according to the traveler), caused by a given magnitude of instantaneous velocity change, is proportional to the separation L of the twins.

In particular, this means that if L = 0 (i.e., if the twins are co-located), the instantaneous velocity change has NO effect on the age of the home twin. That's why the initial and final accelerations by the traveler (in the case of the classic twin "paradox" scenario, where the twins are co-located at the beginning and end of the voyage) don't affect the outcome of the twin "paradox" problem at all ... the classic twin scenario can always be reformulated with no accelerations at the beginning and end of the voyage, without changing the overall outcome.

And because their separation shows up as a proportionality factor in the equation, that means that, for a given velocity change, the effect on the age change of the home twin becomes greater when their separation is larger. In fact, you can see from the equation that the home twin's age can suddenly change by (almost as much as) twice the separation (because delta(v) can have a magnitude as large as 2 minus an infinitesimal amount.).

Here's an example of the use of the equation:

Suppose the twins are 30 lightyears apart at some instant of the traveler's life, and that the traveler's velocity has been constant at v1 = -0.8 for some period of time up until that instant.

The negative sign means that the twins have been moving toward each other. (For simplicity, I'm not bothering to write the units of velocity (lightyears/year)).

Suppose that the traveler then instantaneously changes his velocity to v2 = 0.6. Then

delta(v) = v2 - v1 = (0.6) - (-0.8) = 1.4,

and so we get

delta(CADO_T) = -30 * 1.4 = -42 years.

So, with the given instantaneous velocity change, and with the given separation, the home twin gets YOUNGER by 42 years (according to the traveler), during the traveler's instantaneous change of velocity.

Once you've got the capability to quickly and easily calculate the instantaneous age-changes of the home twin (according to the traveler), caused by instantaneous velocity-changes by the traveler, you can easily get the complete solution for how the home twin's age changes (according to the traveler), during a complete (and fairly complicated) voyage by the traveler, consisting of any arbitrary sequence of instantaneous velocity-changes, separated by periods of constant velocity in between those velocity-changes. The ageing of the home twin, during each of the constant-velocity periods, is easily calculated from the well-known time dilation result, and the home twin's ageing during each of the instantaneous velocity-changes can be easily calculated using the above delta(CADO_T) equation ... the combination of all those amounts of ageing gives the complete solution.

Mike Fontenot

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