How does acceleration affect time dilation in Special Relativity?

[evanlee]

Hi! I'm new to the forum (although I've been reading posts for a while), so this is my first post.

Anyway, I am studying electrical engineering and am taking my second semester of physics currently. I have been taught Special Relativity, and have a good grasp on the basic concepts. In all of the examples we used, and in all of the equations we learned, a constant velocity is assumed. I have been led to believe that acceleration must be answered using General Relativity -- which I really only have a very basic understanding of.

My question is this:

If, say, a spaceship were traveling close to the speed of light, and then had to stop and turn around, how would time dilation be affected? I know that in Special Relativity, from the frame of an observer on Earth, time on the ship must slow down. How then, would a ship, on a round trip, be affected by time dilation? And how can those effects be explained?

Now suppose the ship did not have to stop and turn around, but kept at its constant velocity and turned in a circular pattern. According to classical mechanics, the ship would then deal with an angular acceleration (right?). So then, how would time dilation affect the ship? I have a hunch that this all refers to the fact that the ship, in both cases, must accelerate (or decelerate) to change its direction, but I do not know what this means for time dilation.

Thanks!

 

[JesseM]

Anyway, I am studying electrical engineering and am taking my second semester of physics currently. I have been taught Special Relativity, and have a good grasp on the basic concepts. In all of the examples we used, and in all of the equations we learned, a constant velocity is assumed. I have been led to believe that acceleration must be answered using General Relativity -- which I really only have a very basic understanding of.

No, you don't need general relativity to deal with acceleration, you only need it to deal with gravity. An object of negligible mass accelerating in empty space can be dealt with just fine by SR. See this page, for example.

If, say, a spaceship were traveling close to the speed of light, and then had to stop and turn around, how would time dilation be affected? I know that in Special Relativity, from the frame of an observer on Earth, time on the ship must slow down. How then, would a ship, on a round trip, be affected by time dilation? And how can those effects be explained?

If you look at the ship from the perspective of some inertial frame, then if the ship's instantaneous speed at a particular moment is v, then for some infinitesimal interval of coordinate time dt in that frame, the clock will only advance forward by $$\sqrt{1 - v^2/c^2} * dt$$ in that interval due to time dilation. So, for a clock whose velocity is changing relative to your chosen inertial frame, if you know the speed as a function of coordinate time v(t) in terms of that frame's coordinates, and you want to figure out how much time elapses on the clock between two coordinate times t0 and t1, you evaluate the integral $$\int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dt$$ and that'll give you the right answer (see here).

Now suppose the ship did not have to stop and turn around, but kept at its constant velocity and turned in a circular pattern.

Velocity is a vector, it has both magnitude and direction. If you're moving in a circle, then even if your speed is constant your direction is always changing, so your velocity is not constant--this is a form of acceleration. Just as with linear acceleration, the accelerating observer will know he's accelerating because he'll feel G-forces (the 'centrifugal force' in the case of circular motion). An observer moving at constant velocity in SR will always feel weightless.

According to classical mechanics, the ship would then deal with an angular acceleration (right?). So then, how would time dilation affect the ship? I have a hunch that this all refers to the fact that the ship, in both cases, must accelerate (or decelerate) to change its direction, but I do not know what this means for time dilation.

If you're looking at the inertial frame where the center of the circle is at rest, then the ship is moving at constant speed in this frame, and time dilation is only a function of speed, so in this frame the ship's clock will be running slow by a constant amount. In other inertial frames the speed would be varying, so the rate of clock slowdown would be varying too.

 

[Fredrik]

I have been led to believe that acceleration must be answered using General Relativity

This is one of many things I hate about how SR is being taught. Students always believe this after being exposed to the standard presentation.

That claim is true about the original 1905 version of SR, but it's not true about what physicists today think of as special relativity. The modern version of special relativity includes a few things that were introduced by GR, in particular the manifold structure of the mathematical model of spacetime and a more sophisticated set of postulates that identifies things in the model with things in the real world.

All the other coordinate systems are included as a part of the definition of a manifold. A coordinate system is just a function that maps open subsets of Minkowski space bijectively onto open subsets of $\mathbb R^4$.

A mathematical model on its own can't be a theory of physics. It can't make predictions about the results of experiments until we have postulated a set of identifications between the model and the real world. We can e.g. postulate that a grid of rulers and synchronized clocks (synchronized according to a specific operational definition of simultaneity) would assign coordinates to events in a way that agrees with the assignment of coordinates made by a global inertial frame (which has an abstract mathematical definition). That postulate is sufficient to give us a theory of physics that can handle constant velocity, but insufficient to give us a theory that can handle acceleration.

That's basically were SR was in 1905. It turned out that it wasn't possible to use the same postulate in GR, so it had to be replaced by others, e.g. that what a clock measures is the "proper time" (defined as a certain integral) along the curve in the spacetime manifold that represents its motion. (Another option is to keep talking about that "grid", but postulate that it only assigns coordinates in a way that agrees with the appropriate inertial frame in the limit where the size of the region of spacetime that's being measured goes to zero).

That's really the only difference between the theory that's able to handle acceleration and the one that isn't. The mathematical model is the same (or essentially the same), but if the postulates that identify things you measure with stuff in the model are too naive, the theory will be crippled.

 

[JesseM]

That claim is true about the original 1905 version of SR, but it's not true about what physicists today think of as special relativity. The modern version of special relativity includes a few things that were introduced by GR, in particular the manifold structure of the mathematical model of spacetime and a more sophisticated set of postulates that identifies things in the model with things in the real world.

What do you mean when you say the 1905 version couldn't handle acceleration? You can definitely use 1905 SR to analyze the behavior of an accelerating object from the perspective of an inertial coordinate system--see the time dilation integral I posted in my previous post, or Einstein's comment about treating a curved path as equivalent to a polygonal line near the end of section 4 of the 1905 paper. It is true that the equations of 1905 SR can't be used in an accelerating coordinate system, but that's not the same as saying the theory can't handle acceleration.

 

[Fredrik]

Yes, I expressed myself a bit poorly, but you seem to have understood what I really meant.

 

[gonegahgah]

I've posted some questions in another thread (off topic unfortunately) that are to do with acceleration, gravity and time dilation.
Can anyone in this thread possibly help me further with what the answers should be?
The thread is https://www.physicsforums.com/showthread.php?t=251174&page=5".

 

[mtallingham]

Einstein’s theory of general relativity has been recently proved with the use of supercomputers, but has there been a vital section of the theory left out.
In the question of what binds atoms together to form a molecule and hence matter.
According to measurements the speed of an electron in orbit around a nuclei is slower than or at the speed of light.
As the electron leaves the valance shell of one atom to move to the valance shell of another atom, is there likely to be acceleration, a gravitational pull to get the electron to leave one valance shell to move to another. Is there possibly a combined gravitational pull of the electron and nuclei together to get an electron to leave its orbit and join the orbit of a neighbouring atom?
As it leaves; it accelerates, then once attracted into the orbit of its neighbour, its decelerates and settles into its new orbit, until the combined attraction the neighbouring atom and its orbiting electrons pull it out of orbit again.
In this area of molecular bonding is the energy so intense, with so much excitement, with electrons traveling faster that the speed of light for a short while. Creating a mish-mash of intense energy, that light can’t actually pass through.

It’s like playing football or red rover all over; you can’t get from one end of the field to the other if the other man or men are quicker that you.
Light is traveling along and as it approaches matter, it tries to pass through. Light can pass through oxygen because there is not a huge amount of energy at the changeover point, at the point of bonding. Light can’t can pass through iron, because the energy at the changeover point is intense and creates a barrier to light, that is “faster than the speed of light”. More energy, light can’t get through, partially absorbed, partially reflected, therefore the matter becomes become visible. The amount of energy contained in matter is dependent on how much energy there is at the changeover or bonding point. Light can get through invisible things (oxygen), but can’t get through visible or material things (matter). Hence, according to Einstein; energy in matter is equal to the speed the electrons travel from one orbit to another. Did he realize that if light couldn’t pass through something there must be something in there traveling faster that the speed of light. The more dense the matter, the more energy in it, holding it tighter together. The light absorbed by the solid matter it can’t pass through, buzzes around with the electrons binding the whole thing together in the form of heat.

Valid assumptions or orbiting space junk?

 

[JesseM]

As the electron leaves the valance shell of one atom to move to the valance shell of another atom, is there likely to be acceleration, a gravitational pull to get the electron to leave one valance shell to move to another. Is there possibly a combined gravitational pull of the electron and nuclei together to get an electron to leave its orbit and join the orbit of a neighbouring atom?

The force that binds electrons to atomic nuclei is not gravity but the electromagnetic force. The electromagnetic force between charged objects is vastly stronger than the gravitational force between them, the only reason gravity is more significant at large scales is that large objects (like planets and stars) contain almost exactly equal numbers of positively-charged and negatively-charged particles, so their charge is almost exactly zero.

 

[mtallingham]

Thanks Jesse, Sorry I posted it twice, am a bit new to this


So does the attraction of the neighbouring atom’s electromagnetic force, cause the electron to accelerate out of its orbit into the neighbouring orbit, thus creating a massive amount of energy.

 

[JesseM]

So does the attraction of the neighbouring atom’s electromagnetic force, cause the electron to accelerate out of its orbit into the neighbouring orbit, thus creating a massive amount of energy.

Energy can't be created or destroyed, only transformed from one form to another, in both classical or quantum physics. An electron can gain the energy needed to escape one atomic nuclei if it absorbs a photon with enough energy, for instance. In some cases an electron can be bound to two different nuclei at once, which is the basic idea of how multiple atoms can form chemical bonds.

 

[mtallingham]

So the charge in light negates the charge on the electron and helps it escape its orbit to travel to the neighbouring one. Light contains the necessary energy to facilitate the bond?
Would the electron being measured as being in two nuclei at once be in that moment of time, where it is transiting orbits?

 

[JesseM]

So the charge in light negates the charge on the electron and helps it escape its orbit to travel to the neighbouring one.

No, light has no charge. It's just a matter of light giving enough kinetic energy to the electron to overcome the potential energy which binds the electron to the nucleus.

Would the electron being measured as being in two nuclei at once be in that moment of time, where it is transiting orbits?

In quantum mechanics an electron doesn't have a well-defined path because of the uncertainty principle, which says the more you try to pin down the electron's position the more uncertainty you have about its velocity and vice versa. At any given moment you have a probability distribution for where the electron might be if you try to measure its position, but there's no way to predict it beyond that, and it won't necessarily be at a position between the two nuclei.

 

[mtallingham]

Assuming the path is orbital, until the energy in light due to its motion basically knocks the electrons free from their orbit, letting them transit to another orbit. As the kinetic energy in light presumably is constant, could we determine how much extra energy it adds to the molecular bond to determine total energy in a bond, to determine why there is not enough energy in certain bonds to stop light getting through and why there’s is more energy in other bonds that can stop light getting through, thereby creating solid or visible matter as opposed to gaseous or invisible matter.

 

[JesseM]

Assuming the path is orbital

Like I said, you can't assume that in quantum mechanics. How much do you know about QM? Are you familiar with the uncertainty principle?

until the energy in light due to its motion basically knocks the electrons free from their orbit, letting them transit to another orbit. As the kinetic energy in light presumably is constant

The energy depends on the frequency of the photon, it's given by the equation E = hf, where h is Planck's constant.

could we determine how much extra energy it adds to the molecular bond to determine total energy in a bond

The potential energy is already known.

to determine why there is not enough energy in certain bonds to stop light getting through and why there’s is more energy in other bonds that can stop light getting through, thereby creating solid or visible matter as opposed to gaseous or invisible matter.

I haven't studied this subject, but apparently transparency does depend on the likelihood that passing photons get absorbed by electrons in the material--for instance, http://acept.asu.edu/PiN/rdg/visnxray/visnxray.shtml says:

The absorption of photons in solids depends on the number of electrons that can accept a transfer of energy from the photon. Not all electrons meet this criterion. Some electrons are so tightly bound to their orbital around the atomic nucleus that the photon energy is unable to break the bonds. Other electrons are involved in the bonding between atoms and cannot be freed by the energy of the incident photons. For example, ordinary window glass is transparent to visible photons, but is strongly absorbent to ultraviolet radiation where the photons have energies only a few times greater than that of visible light.

The wikipedia article on absorption in optics also has some basic info.

 

[evanlee]

Thanks everyone for helping me figure out this problem. If I'm reading it correctly, it seems that if an object is accelerating at relativistic velocities, then time dilation (and all other effects) will be varying, just as the velocity is. That does make sense! Thanks again!

 

[mtallingham]

I don’t know enough QM. The energy is defined as a force by a constant. That’s what I’m driving at. I think electrons must move at different speeds, slower in invisible matter, much faster in solid matter. The faster the electron goes, the more dense the matter, the harder for light to get through and get the electron out of orbit thereby creating a series of random bonds and thereby creating molecules and matter. Regardless of how light achieves it purpose, are we able to actual examine bonds to measure energy in them or even manipulate them.

 

[JesseM]

The energy is defined as a force by a constant.

Energy has units of force * distance. For example, if you want to move an object a certain distance in a field of constant force, it will take an amount of energy equal to the force times the distance.

I think electrons must move at different speeds, slower in invisible matter, much faster in solid matter.

Why do you say that? Keep in mind that while kinetic energy increases with speed, potential energy does not depend on speed, only on position (in the case of electrons, it would depend on distance from the nucleus)--two particles at the same position with different speeds would have the same potential energy.

 

[mtallingham]

We must be thinking the same thing. The speed of light. 186.000 miles per second. Is it because we are on Earth and its 186 million miles across the diameter of the orbit and light takes a thousand seconds to cross it.
The speed of the Earth in its orbit comes is 18.6 miles a second, I think. The comparison of electrons around atoms as to planets around suns or stars is universal, but what of the speed of Venus and Mars in their orbits around our sun.
That’s what I’m wondering. Does the distance from the nuclei to the orbiting electron determine the speed the electron travels. Maybe iron electrons are closer to the nuclei, hence those electrons move faster; and forge tighter bonds when they jump orbit, more energy, denser matter, harder for light, traveling at our constant speed, to penetrate, using it’s photons to activate electrons and create molecular bonding.

I think electrons in solid matter, when they jump orbits, must accelerate past the speed of light to make it to the new orbit and then decelerate to join the new orbit. In the bond there’s so much energy and high speed movement that light can’t get through.

That’s what Einstein must have meant, the energy in matter is based on the speed of light as the constant and the electrons are exceeding it to make the bond.

Does this make sense?

 

[JesseM]

Does this make sense?

I can't really make sense of any of the things you say in that post, or what reasoning would lead you to say them. But I can assure you that Einstein did not believe the electrons (or any other object) would ever exceed the speed of light.

 

[mtallingham]

Is there a scale that measures the distance of the electon orbit from the nucleus.

 

[JesseM]

Is there a scale that measures the distance of the electon orbit from the nucleus.

What do you mean by "scale"? The distance of each orbital is measured in ordinary distance units like picometres or Angstroms, and each orbital has a characteristic potential energy measured in units of energy like electron volts. You can see the probability distributions for the electrons to be found at different radii from the hydrogen nucleus in fig. 3-5 and fig. 3-8 on this page.

 

[mtallingham]

This would be why, when light passes though a hydrogen bonded molecule superluminal radiation occurs. General relatively says that a particle traveling at subluminal velocity needs infinite energy to accelerate to the speed of light. No one is talking about accelerating a particle to the speed of light to see how much energy is released in real terms. What it really means is molecular bonding occurs because electrons can escape orbits, transverse orbits, enter new orbits and go back to their old ones. They escape with the help of light. External radiation striking an electron adds enough energy, infinite means exactly that unlimited; the symbol is even the path of an electron orbiting two atoms. Obviously the attraction of the electron to the photon causes it to accelerate to the speed of light, thereby escaping the gravity of one atom. It has nowhere to go until the magnetic field of the neighbouring atom pulls it in. As it is traveling at the speed of light as it crosses the void it must either decelerate to join the orbit of the neighbouring atom or accelerate faster than the speed of light to get into its new orbit. I can’t see how it could decelerate, it would be pulled into the new orbit with such force, that's when it must exceed the speed of light.

 

[Dale]

Thanks everyone for helping me figure out this problem. If I'm reading it correctly, it seems that if an object is accelerating at relativistic velocities, then time dilation (and all other effects) will be varying, just as the velocity is. That does make sense! Thanks again!

Hi evanlee,

This is called http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Clock_Hypothesis" (that time dilation is a function of speed only and not acceleration or higher derivatives), and it has been experimentally verified for accelerations up to 10¹⁸ g.

 

[JesseM]

This would be why, when light passes though a hydrogen bonded molecule superluminal radiation occurs. General relatively says that a particle traveling at subluminal velocity needs infinite energy to accelerate to the speed of light. No one is talking about accelerating a particle to the speed of light to see how much energy is released in real terms. What it really means is molecular bonding occurs because electrons can escape orbits, transverse orbits, enter new orbits and go back to their old ones. They escape with the help of light. External radiation striking an electron adds enough energy, infinite means exactly that unlimited; the symbol is even the path of an electron orbiting two atoms. Obviously the attraction of the electron to the photon causes it to accelerate to the speed of light, thereby escaping the gravity of one atom. It has nowhere to go until the magnetic field of the neighbouring atom pulls it in. As it is traveling at the speed of light as it crosses the void it must either decelerate to join the orbit of the neighbouring atom or accelerate faster than the speed of light to get into its new orbit. I can’t see how it could decelerate, it would be pulled into the new orbit with such force, that's when it must exceed the speed of light.

Your ideas are completely wrong according to current quantum theory, and quantum theory can explain everything about the behavior of electrons and photons without the need for superluminal speeds or infinite energy. Perhaps you should learn a little bit about the mainstream theory, and why physicists think the evidence for it is so persuasive, before declaring it wrong and making up your own non-quantitative alternative.