Special and General Relativity Special and General Relativity Physics Help Forum

Jan 24th 2014, 06:15 PM   #1
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 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 in the 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 the traveler's life, according to the home twin, and the asterisk denotes multiplication.
However, I just realized that the dimensions are wrong!!

CADO_H (time) = CADO_T (time) - v ( meters/second) * L (metres)

"meters / second * meters" does not equal "time"

How can this be?

 Jan 24th 2014, 10:41 PM #2 Senior Member   Join Date: Nov 2013 Location: New Zealand Posts: 552 ok, I figured one of my mistakes. The expression is more correctly phrased as: CADO_T = CADO_H - v * L / c^2 This corrects the dimensions. Also I noticed that (L/c) is the time taken for a light pulse to leave the home twin and arrive at the travelling twin. The (v/c) is a dimensionless fraction of the speed of light. What I haven't quite figured out is how they turned the lorentz factor into a linear term in the CADO derivation. Any thoughts?
 Jan 24th 2014, 11:19 PM #3 Senior Member   Join Date: Nov 2013 Location: New Zealand Posts: 552 Also what if the travelling twin travels away at the speed of light. Then after 1 second (assuming our units are in light seconds) we have: CADO_H = CADO_T - 1 (light second/second) * 1 (light second distance) Then CADO_H - CADO_T = -1 Has travelling twin just travelled into home twin's past???
Jan 25th 2014, 07:24 AM   #4
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 Originally Posted by kiwiheretic [...] What I haven't quite figured out is how they turned the lorentz factor into a linear term in the CADO derivation. Any thoughts?
Based on your previous comments, I THINK by "Lorentz factor" you mean gamma. If so, I don't understand your question ... gamma doesn't explicitly show up in that equation. (Gamma IS used, though, in simple cases for easily determining the quantity CADO_H).

Beware that in the your first and third rendering of the CADO equation, you've switched the quantities CADO_T and CADO_H. CADO_T is the quantity that you are trying to to determine: it is the age of the home twin, ACCORDING TO THE TRAVELER, when the traveler's age is t. CADO_H is the (relatively easy-to-determine) age of the home twin, ACCORDING TO THE HOME TWIN, when the traveler's age is t. Note also that ALL of the quantities in the CADO equation are functions of the traveler's age t ... t is the independent variable in the equation. Also note that all of the quantities on the right-hand-side of the CADO equation are from the perspective of the home twin (which are relatively easy to determine).

I think the best use of your time now is to look carefully at the example on the webpage that is worked out for the simple case of an instantaneous turnaround by the traveler, until you're sure you understand that well. THEN, spend some quality time looking carefully at the section of the webpage on the graphical interpretation of the CADO reference frame. I think once you understand that well, a lot of your quandaries above will go away.
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Last edited by MikeFontenot; Jan 25th 2014 at 07:26 AM.

Jan 25th 2014, 05:45 PM   #5
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 Originally Posted by kiwiheretic Also what if the travelling twin travels away at the speed of light. Then after 1 second (assuming our units are in light seconds) we have: CADO_H = CADO_T - 1 (light second/second) * 1 (light second distance) Then CADO_H - CADO_T = -1 Has travelling twin just travelled into home twin's past???
Hold your horses here ! Matter cannot travel at the speed of light since it would
have to be given infinite mass. You should look at some conventional special
relativity.

Jan 25th 2014, 05:55 PM   #6
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Jan 26th 2014, 07:39 AM   #7
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 Originally Posted by Pmb What's the "CADO Equation"?
See posting #2 in the "Some Confusion about the Lorentz Factor" thread.
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Jan 26th 2014, 09:07 AM   #8
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 Originally Posted by MikeFontenot See posting #2 in the "Some Confusion about the Lorentz
Mike - Let me give you some advice. In the future When someone posts a question as simple and as easy to answer as I just did, i.e. What's the "CADO Equation"? I recommend that you do one of three things.

1) State that the equation is the one given in post #1, i.e.
 Originally Posted by kiwiheretic CADO_H (time) = CADO_T (time) - v ( meters/second) * L (metres) "meters / second * meters" does not equal "time"
2) Write down the expression of the CADO Equation

3) Post a link to a web page where the CADO Equation is written and/or defined.

Please don't direct me to a page where I need to read an two rather long paragraphs which contain nothing I need to or want to know. All you had to do was write down this CADO equation that you defined and that was it. I.e. you should merely and simply posted the following;
 The CADO equation is CADO_T = CADO_H - v * L where CADO_T denotes the traveler's conclusion about the home-twin's age, when the traveler's age is t. CADO_H denotes the home-twin's conclusion about the home-twin's age, when the traveler's age is t. v is their current relative speed, according to the home-twin, at the given instant t in the 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 the traveler's life, according to the home twin (the asterisk denotes multiplication.) For details please see https://sites.google.com/site/cadoeq...eference-frame
Then I wouldn't have had to find the thread you referred me to, find the post in the thread that you referred me to and then read it, the first two paragrpahs of which are useless to me since I didn't have to know them to find the definition of the equation.

For the life of me I can't figure out why you responded as you did other than not wanting to be as thorough as I recommend you be.

Last edited by Pmb; Jan 26th 2014 at 09:18 AM.

Jan 26th 2014, 08:25 PM   #9
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 Originally Posted by MikeFontenot I think the best use of your time now is to look carefully at the example on the webpage that is worked out for the simple case of an instantaneous turnaround by the traveler, until you're sure you understand that well. THEN, spend some quality time looking carefully at the section of the webpage on the graphical interpretation of the CADO reference frame. I think once you understand that well, a lot of your quandaries above will go away.
OK, Yep, that's what I'm trying to do now.

If we look at

again I noticed that any time v = 0 that CADO_T = CADO_H

In the example given on the website the traveler twin ages 40 years
while the home twin ages 80 years based on v = 0.866 ly/yr and then an instantaneous turnaround at the end (after 20 years) and a return at the same speed.

However what if the traveler turns off the engines (assuming instantaneous deaceleration here, ie v = 0 in no time) and takes a coffee break at the end of his journey in one direction? Do not the two twins assume the same age at that point? Then his twin would only be 50 years for his 60 years for the sake of a 5 minute coffee break???

 Jan 26th 2014, 09:38 PM #10 Senior Member   Join Date: Nov 2013 Location: New Zealand Posts: 552 Ok, I tried to work it out using a spreadsheet. Code: T (Yrs) V(ly/yr)Gamma L (ly) Delta CT CADO_T CADO_H 0 0 1.0000 0.0000 0.0000 0.0000 0.0000 1 0.1 1.0050 0.1005 -0.0101 -0.0101 0.0000 2 0.2 1.0206 0.3046 -0.0305 -0.0405 0.0204 3 0.1 1.0050 0.4051 0.0405 0.0000 0.0405 4 0 1.0000 0.4051 0.0405 0.0405 0.0405 5 -0.1 1.0050 0.3046 0.0305 0.0710 0.0405 6 -0.2 1.0206 0.1005 0.0101 0.0810 0.0609 7 -0.1 1.0050 0.0000 0.0000 0.0810 0.0810 8 0 1.0000 0.0000 0.0000 0.0810 0.0810 Columns calculated as follows: T = Travellers time in years Gamma = 1 / Sqrt( 1 - v*v) L = v * Gamma (because its according to home twin) Delta CADO_T (Delta_CT) (row) = -L * (v[row] - v[row-1]) CADO_T (row) = CADO_T(row - 1) + Delta CADO (row) CADO_H = CADO_T + v * L What's wrong with these results? They don't look right. Last edited by kiwiheretic; Jan 26th 2014 at 11:48 PM. Reason: Fix wording