Twins paradox and other special relativity topic discussion

[Qfwfq]

Folks the twin paradox is not really a problem in SR. it poses no logical contradiction because there's no objective comparison between their elapsed proper times if neither one accelerates according to inertial coordinates. It's a much more tricky issue in GR.

 

Only one twin was given energy relative to the other twin. Only one twin has the energy for real relativistic affects. Only the one with energy will show any real time dilation.

This is not an argument according to SR, it is in conflict with the principle of relativity. The only distinction is between inertial and non-inertial coordinates.

 

Each of them would see the other age more, but in fact they would arrive home being the same age.

No, they can have elapsed different proper times, according to how much each of their paths differ from being straight in inertial coordinates.

 

Special relativity is overly simplistic becouse it doesnt take accelerations into account.

It does. Lorentz covariant dynamics describe how a particle will move subject to a force. The best way of working things out is with four-vectors. Consider the following definitions:

 

[math](d\tau)^2=(dx_0)^2-(dx_1)^2-(dx_2)^2-(dx_3)^2[/math] (square of proper time)

 

[math]u_i=\frac{dx_i}{d\tau}[/math] (four-velocity, with the property that [math]u^i u_i=1[/math])

 

[math]f_i=m\frac{du_i}{d\tau}=m\frac{d^2 x_i}{d\tau^2}[/math] (four-force, equal to mass times four-acceleration)

[modest]

Folks the twin paradox is not really a problem in SR. it poses no logical contradiction because there's no objective comparison between their elapsed proper times if neither one accelerates according to inertial coordinates. It's a much more tricky issue in GR.

 

There's a very good faq-type treatment of the clock hypothesis (for someone at my introductory level of physics in any case) here:

 

Does a clock's acceleration affect its timing rate?

 

quoting a bit:

 

The clock postulate generalises this to say that even when the moving clock accelerates, the ratio of the rate of our clocks compared to its rate is still the above quantity [the gamma factor]. That is, it only depends on v, and does not depend on any derivatives of v, such as acceleration...

 

The clock postulate is one of the ideas we use to build the theory of gravity known as general relativity. People sometimes think (and books sometimes say) that a proper description of accelerating clocks demands general relativity. That's not true at all. General relativity is built on a foundation that includes the clock postulate. Once we've built general relativity, we cannot then turn around and start using it to "explain" the physics of accelerating clocks! That would be circular reasoning. General relativity is built upon the ideas of special relativity, but the physics of accelerating clocks is completely handled by special relativity alone...

 

It then goes on to generalize the clock hypothesis to non-inertial frames.

 

I'm curious then Qfwfq, is it necessary to add the caveat about neither twin accelerating according to inertial coordinates in your quote above? I have enormous respect for your talents on this subject and don't mean to challenge what you've said (besides which, I completely agree with your conclusion), I'm just curious and trying to understand.

 

I also think it might be of some interest for those insisting SR is inadequate for the twin scenario that Einstein introduced the topic in his original 1905 paper on SR. A significant portion of section 1.4 is dedicated to reunited clocks that are no longer synchronized.

 

If we assume that the result proved for a polygonal line is also valid for a continuously curved line, we arrive at this result: If one of two synchronous clocks at A is moved in a closed curve with constant velocity until it returns to A, the journey lasting t seconds, then by the clock which has remained at rest the travelled clock on its arrival at A will be 1/2tv^2/c^2 second slow.

 

~modest

[Qfwfq]

is it necessary to add the caveat about neither twin accelerating according to inertial coordinates in your quote above?

SR means that spacetime is considered flat, actually this doesn't quite suffice to imply it not having an "odd" global topology but typically this is implicitly assumed, with coordinate space being [imath]\mathbb{R}^4[/imath]. With such assumptions, if neither clock is accelerated then their world lines can at most intersect in one single point. Does this help?

 

It implies there being no comparison between elapsed proper times that isn't arbitrary. What intervals can actually be compared, independently of coordinate choice? Between which two points? The only "objective" way of defining a mapping between the world lines is in terms of proper time anyway, so the point becomes moot.

 

If OTOH there are at least two intersections of their respective world lines then the elapsed proper time between a pair of these intersections, for each world line, is definable independently and yet is a scalar and so is the difference (which might not be zero). There is an actual epistemological stump if and only if spacetime be such that even distinct geodesics can have more than one intersection.

[modest]

Thank you Q. The "no objective comparison" part is exactly what I didn't follow. In fact, I'm sure now I read it wrong :hihi:

 

~modest