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Relativity And Simple Algebra

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#477 ralfcis

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Posted 17 March 2019 - 07:21 AM

Yes and then time evolved to using telegrams to relay reality to us all. Much faster that way. The only thing I can't yet explain is why your brain hasn't yet got the message. Good questions though, they keep me on my toes and make me try to explain things in terms of what a very young child might understand but, obviously, I'm not quite there yet.


Edited by ralfcis, 17 March 2019 - 07:42 AM.


#478 ralfcis

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Posted Yesterday, 02:53 PM

I've been away for a while because I've had a devil of a time with the math. This was from a few posts back to prove if Alice chenges her velocity from .6c to c away from Bob, she will age 6 yrs from 4 to 10 while Bob will only age 4 yrs from 4 to 8 during the time of relative velocity imbalance. It turns out my initial equation for vwas about .5 c while it's the slope of the line of perspective simultaneity of slope 1/vthat is the line which intersects the Alice's velocity line c. So let's start over:

 

Correction: my new formulas are ( x-3 ) /t = v/c and  x / ( t - 3 )  = c/vh

 

I don't quite understand why each velocity is divided by c but it doesn't affect the numbers and makes the math really crisp. The time results end up without units though.

 

x = (t-3) / (vh/c)

plugging in x = vt/c + 3

 

we get (t-3) / (vh/c) = vt/c + 3

 

t(1-vvh/c2) = 3(vh/c + 1)

 

so t = 3/c (vh + c)/ ( 1 - vvh/c2)

 

knowing t'=t/Y and plugging in Y = (c2 + vh2) / (c2 - vh2) and v = 2 c2 vh/ (c2 + vh2)

 

t' = 3/c (vh + c)(c2-vh2)/ ((( 1 - 2vh2c2 / (c2(c2+vh2))(c2+vh2))

   = 3/c (vh + c)(c2-vh2)/ (c2+vh-2vh2))

 The term (c2-vh) that was causing infinity gets cancelled out so

 

we get t' = 3(vh + c) /c

 

Bob's age is 8 and Alice's age is 4 + t'. So as v-> c, t' =6

QED

 

Here are some more numbers to play around with in the last equation

 

v c      vh c   Age diff (Alice ends up older by)

1         1       2

40/41    4/5      1.4   

15/17    3/5      .8

4/5        1/2      .5

3/5        1/3      0

 

Now we can continue the math opening up new velocities of relativity's landscape that are forbidden by relativity to determine age difference. 


Edited by ralfcis, Today, 06:34 AM.


#479 ralfcis

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Posted Yesterday, 06:36 PM

I forgot I was supposed to derive the equation:

 

(Y+ 1) = 1 / (1 - vh/v)

 

Solving for Y we get

 

Y = vh/v / (1 - vh/v) or  Y = 1 / (v/vh - 1)

 

I just stumbled on these equations and they turn normal equations with square roots and squares into simple fractions. The derivation is steered by knowing the final result in advance.

 

Start with 

 

Y = (c2 + vh2) / (c2 - vh2)

 

 = vh (c2   + vh2) / ((2c2 - c2 - vh2) vh )  

 

 = vh / ((2c2vh - vhc2 - vh3) / (c2 + vh2))

 

 Knowing v = 2 c2 vh/ (c2 + vh2)

 

Y = vh/ ( v - vh)

 

So Y = 1 / (v/vh - 1)

and Y = vh/v / (1 - vh/v)

and (Y+ 1) = 1 / (1 - vh/v)

QED

 

This shows the mathematical diversity of just using v and vh in equations instead of squares and roots.


Edited by ralfcis, Yesterday, 09:33 PM.


#480 ralfcis

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Posted Yesterday, 08:33 PM

we derived t' = 3(vh + c) /c

 

3 is the separation in ly between Bob and Alice when she makes a change in velocity. Let's call it X for separation at turning point.

 

So t' = X(vh + c) /c

 

This equation is now more general. In this example of X=3, we've seen a table predicting Alice's age difference with Bob based on her velocity choices away from Bob:

 

v c      vc   Age diff (Alice ends up older by)

1         1       2

40/41    4/5      1.4   

15/17    3/5      .8

4/5        1/2      .5

3/5        1/3      0

 

For the same velocity choices towards Bob, we get a similar table:

 

v c      vc   Age diff (Alice ends up younger by)

1         1       2+2

40/41    4/5      2+1.4   

15/17    3/5      2+.8

4/5        1/2      2+.5

3/5        1/3      2+0

 

Both tables show a range from .6c to c. But what about changes from .6c to 0c? vx = 0c is equivalent to vt =c. vx = .6c is equivalent to vt =.8c. This is where we take advantage of the equation v2 = vx2 + vtand see if the pattern in age difference continues between the range of .6c to 0c to -.6c.

 

Here is a table comparing vx and v (vt = c/Y,  Y = 1 / (v/v-1), v = vh (vt +1))

 

vx c     vhx c     vt c     vht  c   Proposed age diff pattern

 

 

1         1           0         0

40/41  4/5        9/41    1/9

12/13  2/3        5/13    1/5

15/17  3/5        8/17    1/4

4/5      1/2        3/5      1/3     

3/5      1/3        4/5      1/2      y+ 0

8/17    1/4        15/17  3/5      y+.5

5/13    1/5        12/13  2/3      y+.8

9/41    1/9        40/41  4/5      y+1.4

0          0          1         1         y+2

 

This table should allow us to use the age difference numbers for vx from 3/5 c to c for age difference numbers from 3/5 c to 0c in reverse order. I'll need some time to verify this pattern.

 

PS. The equation v = vh (v/c +1gets derived like this:

 

Y = vh / (v - vh)

Yv -Yvh = vh

vt = c/Y 

cv/vt -cvh/vt =vh

v - vh = vhvt / c

v = vh ( vt/c + 1)


Edited by ralfcis, Today, 06:48 AM.


#481 ralfcis

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Posted Today, 06:11 AM

If all this works out, it will show the perfect symmetry between velocity through space and velocity through time. Einstein made the assumption that the symmetry was between space and time itself so he mashed the two together into spacetime and made time into just another space dimension but it acts nothing like a space dimension. As a result of this mashup, both time and space conspire to keep c constant from all perspectives. All the math I've presented here, which explains all relativistic phenomena without contradiction, keeps space and time separate without any need for length contraction. Of course the only feedback I get is that it's all so wrong, no one even knows where to begin. They're right, no one even knows where to begin because no one knows very much. Both extremes in this debate, the relativists and absolutists, have their eyes shut tight. No way to reach them.