#### Mike_Fontenot

**Registered Senior Member**

First, here is an excerpt, from Section 2 of my webpage, of the

____________________________

Given the above definition of the CADO frame, it is possible to derive a very simple, and very useful equation, called "the CADO equation",[5] which allows the traveler to determine, at each instant

First, it is important to understand that, for any given instant

The CADO equation can be written (most simply) as

where

the traveler's life, with

traveler's life, according to the home twin,

and

the asterisk denotes multiplication.

Strictly speaking, the quantity

The above equation gives the relationship between those four quantities (

What makes the CADO equation especially useful is that it allows the quantity

_____________________________________

I originally derived the CADO equation, many years ago, from the Minkowski diagram. I have recently made an addition to the webpage, to explicitly show how to do that derivation. The addition is near the end of Section 11 of the webpage (the section entitled "Graphical Interpretation of the CADO Frame).

Here is a link to the webpage:

https://sites.google.com/site/cadoequation/cado-reference-frame

*definition*of the CADO equation:____________________________

Given the above definition of the CADO frame, it is possible to derive a very simple, and very useful equation, called "the CADO equation",[5] which allows the traveler to determine, at each instant

*t*in his life, the current age of any given distant perpetually-inertial object or person (the "home-twin" in the twin paradox scenario).First, it is important to understand that, for any given instant

*t*in the traveler's life, the home-twin and the traveler will generally*disagree*with one another about how old the home-twin is at that instant of the traveler's life. There are two quantities in the CADO equation which represent each of the twins' conclusions about the home-twin's age when the traveler's age is*t*. The quantity*CADO_T*denotes the*traveler's conclusion*about the home-twin's age, when the traveler's age is*t*, whereas the quantity*CADO_H*denotes the*home-twin's conclusion*about the home-twin's age, when the traveler's age is*t*.The CADO equation can be written (most simply) as

*CADO_T*=*CADO_H*-*v***L*where

*v*is their current relative speed, according to the home-twin, at the given instant*t*inthe traveler's life, with

*v*taken as positive when the twins are moving apart,*L*is the distance from the home-twin to the traveler, at the given instant*t*in thetraveler's life, according to the home twin,

and

the asterisk denotes multiplication.

Strictly speaking, the quantity

*L(t)*is the position of the traveler, relative to the home-twin, according to the home-twin, when the traveler's age is*t*. The distinction will be clarified later (in Section 5), but for now, it's simplest to just think of it as a distance (a number either positive or zero).The above equation gives the relationship between those four quantities (

*CADO_T*,*CADO_H*,*v*, and*L*), at the given instant*t*of the traveler's life. I.e., although it is not shown explicitly, each of the four quantities in the equation are functions of*t*.What makes the CADO equation especially useful is that it allows the quantity

*CADO_T*, which is a quantity which is otherwise relatively difficult to determine, to be easily calculated from the other three quantities (*CADO_H*,*L*, and*v*), which are each very easy to determine._____________________________________

I originally derived the CADO equation, many years ago, from the Minkowski diagram. I have recently made an addition to the webpage, to explicitly show how to do that derivation. The addition is near the end of Section 11 of the webpage (the section entitled "Graphical Interpretation of the CADO Frame).

Here is a link to the webpage:

https://sites.google.com/site/cadoequation/cado-reference-frame

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