# Two questions regarding special relativity and acceleration

#### MontBlanc2012

##### Member
I appreciate your input, but really, I'm not doing science here. I'm not nearly qualified for that. All I want to do is throw a bit of code together for a fun time...
Sure - enjoy.

• jedidia

#### Linguofreak

##### Well-known member
The use of special relativity to address question relating to accelerating reference frames relies on the Clock Hypothesis - namely that “when a clock is accelerated the effects of the motion on the rate of the clock is no more that associated with its instantaneous velocity - the acceleration adds nothing.”

This question of the validity of the Clock Hypothesis remains an open issue. If the Clock Hypothesis holds, then it is entirely appropriate to use special relativity to address jedidia’s questions. If it does not, then you might need to think again.

The clock hypothesis is an unavoidable consequence of the geometry of Minkowsky spacetime. If it does not hold physically, then Special Relativity is not a description of local physics, nor is General Relativity a description of the physics of the universe as a whole.

Frankly, I'm not sure if you could devise a consistent geometry in which the Clock Hypothesis wouldn't hold. That said, I'm not sure that you, or the author of the paper you linked, understand what the clock hypothesis is actually saying. Time dilation depends directly only on instantaneous velocity and not on acceleration, but the evolution of instantaneous velocity depends on acceleration, and so if we have a set of three events and two worldlines that both visit each of the three events, but with different acceleration profiles, they will necessarily be different worldlines and will, unless carefully chosen, develop different proper times.

Consider the Euclidean equivalent of the twin paradox, you have two parallel lines, A and B. You have a car parked on line A, paralell to it, at point a1. Your coordinates at any point are x, y, and your total distance driven is D. You start driving, turn toward line B, touch it, and return to line A, facing parallel to the line, at point a2. D, integrated along the path you drove, is different than the straight line distance from a1 to a2. This is equivalent to the different ages of the twins. At any given point along your path dy/dD depends only on your present heading, and not on how tightly you're turning at that point, this is equivalent to the clock hypothesis. But depending on what turns you make and how tight they are, the total D along your path will be different, this is equivalent to the dependency of proper time on your acceleration profile.

But to deal with the effect of the curvature of your path on the total distance traveled, you don't need to know how to deal with the effect of hills or the curvature of the Earth on the geometry of the surface you're driving on unless there is actually a hill involved, or the distance between lines A and B is an appreciable fraction of the Earth's circumference. This is equivalent to the fact that you don't need GR to deal with accelerations in relativity.

When you return from the Euclidean model to SR, all that you're changing is that you flip the sign of the y^2 term in the Pythagorean theorem and relabeling y as "t".

The equivalence principle, of course, lies at the heart of general relativity - and no doubt Einstein developed the general theory because he recognised that the special theory was not sufficient to house the concept.

Yes, but he came at it from the opposite direction to what you're implying: Newtonian gravitation was incompatible with SR. Einstein knew how to deal with acceleration in SR, and he used that and the equivalence principle to deduce what sort of gravitational theory would be compatible with SR. If the way he was handling accelerations in SR was invalid, then the entire foundation of GR collapses.

#### MontBlanc2012

##### Member
The clock hypothesis is an unavoidable consequence of the geometry of Minkowsky spacetime. If it does not hold physically, then Special Relativity is not a description of local physics, nor is General Relativity a description of the physics of the universe as a whole.

Frankly, I'm not sure if you could devise a consistent geometry in which the Clock Hypothesis wouldn't hold. That said, I'm not sure that you, or the author of the paper you linked, understand what the clock hypothesis is actually saying. Time dilation depends directly only on instantaneous velocity and not on acceleration, but the evolution of instantaneous velocity depends on acceleration, and so if we have a set of three events and two worldlines that both visit each of the three events, but with different acceleration profiles, they will necessarily be different worldlines and will, unless carefully chosen, develop different proper times.

Consider the Euclidean equivalent of the twin paradox, you have two parallel lines, A and B. You have a car parked on line A, paralell to it, at point a1. Your coordinates at any point are x, y, and your total distance driven is D. You start driving, turn toward line B, touch it, and return to line A, facing parallel to the line, at point a2. D, integrated along the path you drove, is different than the straight line distance from a1 to a2. This is equivalent to the different ages of the twins. At any given point along your path dy/dD depends only on your present heading, and not on how tightly you're turning at that point, this is equivalent to the clock hypothesis. But depending on what turns you make and how tight they are, the total D along your path will be different, this is equivalent to the dependency of proper time on your acceleration profile.

But to deal with the effect of the curvature of your path on the total distance traveled, you don't need to know how to deal with the effect of hills or the curvature of the Earth on the geometry of the surface you're driving on unless there is actually a hill involved, or the distance between lines A and B is an appreciable fraction of the Earth's circumference. This is equivalent to the fact that you don't need GR to deal with accelerations in relativity.

When you return from the Euclidean model to SR, all that you're changing is that you flip the sign of the y^2 term in the Pythagorean theorem and relabeling y as "t".

Yes, but he came at it from the opposite direction to what you're implying: Newtonian gravitation was incompatible with SR. Einstein knew how to deal with acceleration in SR, and he used that and the equivalence principle to deduce what sort of gravitational theory would be compatible with SR. If the way he was handling accelerations in SR was invalid, then the entire foundation of GR collapses.
Sigh

Enough of this, methinks. Over the years, I’ve read too many differing views of what Einstein actually thought, what Minkowski actually proved, and assorted claims of resolution of the Twin Paradox. You clearly have your views of the universal validity of the Clock Hypothesis. I have my doubts. Let’s let the issue rest there.

#### Linguofreak

##### Well-known member
Sigh

Enough of this, methinks. Over the years, I’ve read too many differing views of what Einstein actually thought, what Minkowski actually proved, and assorted claims of resolution of the Twin Paradox. You clearly have your views of the universal validity of the Clock Hypothesis. I have my doubts. Let’s let the issue rest there.

Let's put it this way: the clock hypothesis definitely holds for special relativity as formulated by Einstein, in which the metric s^2 = t^2-x^2-y^2-z^2 holds. If the criticism of the Kundig and Bailey experiments in the paper linked holds then a different special theory holds, in which the clock hypothesis does not hold, and by the equivalence principle that leads to a different general theory.

#### MontBlanc2012

##### Member
Let's put it this way: the clock hypothesis definitely holds for special relativity as formulated by Einstein, in which the metric s^2 = t^2-x^2-y^2-z^2 holds. If the criticism of the Kundig and Bailey experiments in the paper linked holds then a different special theory holds, in which the clock hypothesis does not hold, and by the equivalence principle that leads to a different general theory.
Yes, that’s about the size of it.