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Re: Simplified Twin Paradox Resolution.
Posted:
Jan 6, 2013 6:43 PM
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On 7/01/2013 1:25 AM, Vilas Tamhane wrote:
> On Jan 6, 11:23 am, Sylvia Else <syl...@not.at.this.address> wrote:
>> On 6/01/2013 3:59 PM, Koobee Wublee wrote:
>>
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>>> On Jan 5, 5:57 pm, Sylvia Else wrote:
>>>> On 5/01/2013 5:59 AM, Koobee Wublee wrote:
>>
>>>>> Instead of v, let?s say (B = v / c) for simplicity. The earth is
>>>>> Point #0, outbound spacecraft is Point #1, and inbound spacecraft is
>>>>> Point #2.
>>
>>>>> According to the Lorentz transform, relative speeds are:
>>
>>>>> ** B_00^2 = 0, speed of #0 as observed by #0
>>>>> ** B_01^2 = B^2, speed of #1 as observed by #0
>>>>> ** B_02^2 = B^2, speed of #2 as observed by #0
>>
>>>>> ** B_10^2 = B^2, speed of #0 as observed by #1
>>>>> ** B_11^2 = 0, speed of #1 as observed by #1
>>>>> ** B_12^2 = 4 B^2 / (1 ? B^2), speed of #2 as observed by #1
>>
>>>>> ** B_20^2 = B^2, speed of #0 as observed by #2
>>>>> ** B_21^2 = 4 B^2 / (1 ? B^2), speed of #1 as observed by #2
>>>>> ** B_22^2 = 0, speed of #2 as observed by #2
>>
>>>>> When Point #0 is observed by all, the Minkowski spacetime (divided by
>>>>> c^2) is:
>>
>>>>> ** dt_00^2 (1 ? B_00^2) = dt_10^2 (1 ? B_10^2) = dt_20^2 (1 ? B_20^2)
>>
>>>>> When Point #1 is observed by all, the Minkowski spacetime (divided by
>>>>> c^2) is:
>>
>>>>> ** dt_01^2 (1 ? B_01^2) = dt_11^2 (1 ? B_11^2) = dt_21^2 (1 ? B_21^2)
>>
>>>>> When Point #2 is observed by all, the Minkowski spacetime (divided by
>>>>> c^2) is:
>>
>>>>> ** dt_02^2 (1 ? B_02^2) = dt_12^2 (1 ? B_12^2) = dt_22^2 (1 ? B_22^2)
>>
>>>>> Where
>>
>>>>> ** dt_00 = Local rate of time flow at Point #0
>>>>> ** dt_01 = Rate of time flow at #1 as observed by #0
>>>>> ** dt_02 = Rate of time flow at #2 as observed by #0
>>
>>>>> ** dt_10 = Rate of time flow at #0 as observed by #1
>>>>> ** dt_11 = Local rate of time flow at Point #1
>>>>> ** dt_12 = Rate of time flow at #2 as observed by #1
>>
>>>>> ** dt_20 = Rate of time flow at #0 as observed by #2
>>>>> ** dt_21 = Rate of time flow at #1 as observed by #2
>>>>> ** dt_22 = Local rate of time flow at Point #2
>>
>>>>> So, with all the pertinent variables identified, the contradiction of
>>>>> the twins? paradox is glaring right at anyone with a thinking brain.
>>>>> <shrug>
>>
>>>> You assert that there are a paradox. I take it you mean in the sense
>>>> that the theory gives two results for one situation, such that they are
>>>> impossible to reconcile.
>>
>>>> I challenge you to show that mathematically, rather than just asserting
>>>> it. Do not just point at the maths above and claim that it's obvious.
>>
>>> PD, are you turning into a troll now? For the n?th time, the
>>> following is one such presentation of mathematics that show the
>>> contradiction in the twins? paradox.
>>
>>> - - -
>>
>>> From the Lorentz transformations, you can write down the following
>>> equation per Minkowski spacetime. Points #1, #2, and #3 are
>>> observers. They are observing the same target.
>>
>>> ** c^2 dt1^2 ? ds1^2 = c^2 dt2^2 ? ds2^2 = c^2 dt3^2 ? ds3^2
>>
>>> Where
>>
>>> ** dt1 = Time flow at Point #1
>>> ** dt2 = Time flow at Point #2
>>> ** dt3 = Time flow at Point #3
>>
>>> ** ds1 = Observed target displacement segment by #1
>>> ** ds2 = Observed target displacement segment by #2
>>> ** ds3 = Observed target displacement segment by #3
>>
>>> The above spacetime equation can also be written as follows.
>>
>>> ** dt1^2 (1 ? B1^2) = dt2^2 (1 ? B2^2) = dt3^2 (1 ? B3^2)
>>
>>> Where
>>
>>> ** B^2 = (ds/dt)^2 / c^2
>>
>>> When #1 is observing #2, the following equation can be deduced from
>>> the equation above.
>>
>>> ** dt1^2 (1 ? B1^2) = dt2^2 . . . (1)
>>
>>> Where
>>
>>> ** B2^2 = 0, #2 is observing itself
>>
>>> Similarly, when #2 is observing #1, the following equation can be
>>> deduced.
>>
>>> ** dt1^2 = dt2^2 (1 ? B2^2) . . . (2)
>>
>>> Where
>>
>>> ** B1^2 = 0, #1 is observing itself
>>
>>> According to relativity, the following must be true.
>>
>>> ** B1^2 = B2^2
>>
>>> Thus, equations (1) and (2) become the following equations
>>> respectively.
>>
>>> ** dt1^2 (1 ? B^2) = dt2^2 . . . (3)
>>> ** dt2^2 = dt1^2 (1 ? B^2) . . . (4)
>>
>>> Where
>>
>>> ** B^2 = B1^2 = B2^2
>>
>>> The only time the equations (3) and (4) can co-exist is...
>>
>> ... never
>>
>> In deriving [1] and [2] you prefaced them with caveats about who is
>> observing whom. So they relate to different measurement situations. You
>> cannot combine them in any meaningful way.
>>
>> Sylvia.
>
> Nobody is observing anybody. Everybody has right to write down
> equations.
>
And then? Koobee is trying to show that the equations contain an
inconsistency, which is to say, that they cannot be a description of any
reality.
Sylvia.
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