Can One Grasp Relativity Without Doing the Math?

[Tumbleweed]

Because geometry is composed of particles, or so it seems. .

Now that's a heavy concept. So it must have a wave function too. Is that why location is just statistical in the quantum world? Does that indicate there are no smooth geometrical lines but quanta lines instead?

 

[sol invictus]

Is that why location is just statistical in the quantum world?

No. Geometry is quantized if gravity is quantized. But even in a world with no gravity at all, particles are probability clouds.

Does that indicate there are no smooth geometrical lines but quanta lines instead?

You can't talk about "lines" at all without having a geometry, so it's kind of worse than that...

 

[Raze]

[...] However Einstein showed there is a way to re-write that law so that it does look the same in all frames, basically by using a set of symbols that transform nicely when you change frames and automatically include all the relevant fictitious forces.

So, my question is this:

Does general relativity or does it not accomplish the same thing for inertial and non-inertial frames that special relativity accomplished for inertial frames?

Or are well still not there yet? Is it not possible?


ETA- Oh, I think I see what you mean. We can describe events in any frame of reference, but locally, say, someone in free fall, will define laws similar to Newton's to describe a ball they give a push to, but someone on the outside will have a different law of motion for the ball? However, both of these descriptions can be formulated and transformed to the other coordinate system with the same mathematical formalism?

Do I get it or am I still lost in the dark?

 

[Raze]

Um, not at all. An accelerating field is the same as a gravitational field a rotating field is much more complex. See Coriolis forces.

You can work out the equations sure, if you know in what way your reference frame is non inertial, but the general form of the equations do hold.

I am not convinced that in an accelerating reference frame momentum would be conserved for example.

Did you mean to say "but the general form of the equations do not hold?"

Because if you meant what you actually posted (that the general form of the equations hold), I would consider that to be equivalent to the statement that the laws of physics hold (of course, that's just me and I could be factually incorrect). ETA- never mind, maybe not. More detail in my response above to Sol.

But if you meant to have a "not" there, then I'd have to say that no, the laws of physics are not the same in every frame, if you are correct.

 

[Fredrik]

So, my question is this:

Does general relativity or does it not accomplish the same thing for inertial and non-inertial frames that special relativity accomplished for inertial frames?

Or are well still not there yet?

General relativity has accomplished that, in exactly the same way that special relativity has. So I'm not sure if I should answer "yes" or "no". It depends on if I can say that special relativity has accomplished...whatever it is you think it has failed to accomplish. Can you be more specific about what problems you think special relativity has with non-inertial frames?

 

[Raze]

General relativity has accomplished that, in exactly the same way that special relativity has. So I'm not sure if I should answer "yes" or "no". It depends on if I can say that special relativity has accomplished...whatever it is you think it has failed to accomplish. Can you be more specific about what problems you think special relativity has with non-inertial frames?

What I mean is that for inertial frames, special relativity gave the correct transformation law that took Maxwell's equations and generalized them to all inertial frames rather than just for the special "ether" frame, and additionally the general laws of mechanics and the law of transformation (conservation of momentum, energy, etc and the Galilean transformation replaced by the Lorentz transformation) were modified where necessary so that the equations that describe them are identical in every inertial frame.

Is this what general relativity does for inertial and non-inertial frames?

ETA- As far as what special relativity failed to accomplish- it doesn't work in non-inertial frames, I presume. For example, a guy on a spinning circle will get a different law for the circle's geometry (a different value of Pi) than a guy standing in an inertial frame outside, because as the guy on the circle measures the circumference of the circle there will be a length contraction factor on his measuring rod, but there won't be when he measures the diameter. (obviously this probably isn't what really happens, but presumably there is a problem with special relativity and non-inertial frames)

 

[Fredrik]

What I mean is that for inertial frames, special relativity gave the correct transformation law that took Maxwell's equations and generalized them to all inertial frames rather than just for the special "ether" frame, and additionally the general laws of mechanics and the law of transformation (conservation of momentum, energy, etc and the Galilean transformation replaced by the Lorentz transformation) were modified where necessary so that the equations that describe them are identical in every inertial frame.

I couldn't find a short way of saying this...

There's something called "general covariance" that holds in both SR and GR. The idea is that the laws of physics are relationships between tensor fields (with spacetime as their domain of definition). These relationships can be expressed without any reference to coordinate systems. So a "law of physics" is actually something coordinate independent.

However, every tensor field has a set of "components" in each coordinate system, and every tensor equation can be expressed in terms of those components instead. There's a rule called "the tensor transformation law" that tells us how to calculate the components of a tensor in an inertial frame, given the components in another inertial frame. That rule is what you use when you "transform" a law of physics from one coordinate system to another. A tensor equation in component form will of course only involve components of the tensors that appear in the coordinate independent equation. The statement that "the laws of physics are the same in all coordinate systems" is just another way of saying that.

In both SR and GR, there's a tensor field that has a very special sigificance, called the metric tensor. Among other things, it tells us which curves in spacetime we should think of as "straight lines", i.e. as representing inertial motion. In GR, the metric tensor is a dynamical quantity, to be dermined by solving an equation. In SR, the metric tensor is fixed. It has 16 components, 10 of which are independent. In a global inertial frame, all but 4 components are 0, and the 4 non-zero components are all 1 or -1.

Suppose that you choose an inertial frame, and express a law of physics that involves the metric tensor in terms of tensor components, and instead of writing e.g. g₂₂ for the "22" component of the metric tensor, you just write 1. Then you simplify the expression as much as you can. Now the equation doesn't look like the original coordinate independent equation. It has taken a "special" form that only holds in global inertial frames.

If you start with a law of physics in that "special" form, and try to use the tensor transformation law to find the components in some other coordinate system, you will usually get the wrong result, because you have to transform the components of the metric tensor too, and if you e.g. see a 2, you don't know if it's =2g₃₃ just the number 2. This won't be a problem if you transform to another inertial frame, since the components of the metric tensor are the same in all of them. So a change from an inertial frame to another (a Lorentz transformation, or more generally, a Poincaré transformation) preserves the "special" form of the equation that only holds in inertial frames. This is called "special covariance". When physicists say that "the laws of physics are the same in all inertial frames", this is what they have in mind.

It should be clear that special covariance is a consequence of general covariance and the fact that the components of the metric tensor are the same in all inertial frames.

Is this what general relativity does for inertial and non-inertial frames?

It wasn't general relativity that solved the problem. It was general covariance, which is now a part of special relativity too. It may not have been a part of SR in 1905, but it's definitely a part of a modern formulation of SR. (It's more or less automatic if you define spacetime as a manifold instead of as a vector space).

ETA- As far as what special relativity failed to accomplish- it doesn't work in non-inertial frames, I presume.

It definitely works. It's just more difficult to do calculations.

 

[ponderingturtle]

Did you mean to say "but the general form of the equations do not hold?"

Because if you meant what you actually posted (that the general form of the equations hold), I would consider that to be equivalent to the statement that the laws of physics hold (of course, that's just me and I could be factually incorrect). ETA- never mind, maybe not. More detail in my response above to Sol.

But if you meant to have a "not" there, then I'd have to say that no, the laws of physics are not the same in every frame, if you are correct.

The thing is that the idea was that no matter the reference frame you could do an experiment and get the same results. THis is simply untrue for non innertial reference frames vs innertial reference frames.

Sure we can come up with equations to account for this, but the experiments would be markedly different.

 

[sol invictus]

So, my question is this:

Does general relativity or does it not accomplish the same thing for inertial and non-inertial frames that special relativity accomplished for inertial frames?

Yes, I'd say it does.

The thing is that the idea was that no matter the reference frame you could do an experiment and get the same results. THis is simply untrue for non innertial reference frames vs innertial reference frames.

Sure we can come up with equations to account for this, but the experiments would be markedly different.

You have to be a bit more careful. What exactly do you mean by "an experiment"? Obviously, some experiments ("what is the speed of object A?") will give different results depending on which frame you're using.

The best way to say it is that relativity - special and general - tells us that physical observables can only depend on coordinate-invariant quantities. An example of a coordinate invariant quantity is the inner product of two 4-vectors (for example the energy-momentum of object A with the energy-momentum of the experimental apparatus). An example of a coordinate non-invariant quantity is a single 4-vector.

Special relativity goes on to assert that the metric of spacetime is the Minkowski metric, which means coordinate-invariant quantities are constructed using that metric. General relativity tells us that the metric in fact isn't Minkowski when gravity is taken into account, and it tells us how to find the metric given a distribution of matter/energy. With those definitions, which I think are the best ones (on the basis of intellectual coherency if not history), special relativity applies perfectly well in non-inertial frames.

 

[ponderingturtle]

You have to be a bit more careful. What exactly do you mean by "an experiment"? Obviously, some experiments ("what is the speed of object A?") will give different results depending on which frame you're using.

I was refering to experiment entirely internal to the reference frame

Special relativity goes on to assert that the metric of spacetime is the Minkowski metric, which means coordinate-invariant quantities are constructed using that metric. General relativity tells us that the metric in fact isn't Minkowski when gravity is taken into account, and it tells us how to find the metric given a distribution of matter/energy. With those definitions, which I think are the best ones (on the basis of intellectual coherency if not history), special relativity applies perfectly well in non-inertial frames.

So how do you conserve momentum in non innertial frames?

 

[sol invictus]

I was refering to experiment entirely internal to the reference frame

I don't know what that means. Every experiment, in fact every thing, is in every reference frame.

So how do you conserve momentum in non innertial frames?

You don't have to do anything - it's just conserved.

What changes is that momentum of an object in a non-inertial frame is not its mass times the derivative of its coordinate location with respect to time. But it's not that in special relativity either.

 

[ponderingturtle]

I don't know what that means. Every experiment, in fact every thing, is in every reference frame.

But the results differ. You can account for them with ficticious forces and the like, but different things happen.

You don't have to do anything - it's just conserved.

What changes is that momentum of an object in a non-inertial frame is not its mass times the derivative of its coordinate location with respect to time. But it's not that in special relativity either.

So there is some prefered reference frame in GR that you measure momentum in? When people talk about conservation laws I always think of it in terms of values and summations. In a non innertial frame I don't see why the summations of different times would have to be equal, because the reference frame is non innertial.

So sure you can postulate a hypothetical innertial reference from and correct the measurements in a non innertial frame to it, but that seems a strange thing to do and still claim that conservation laws hold.

 

[sol invictus]

But the results differ. You can account for them with ficticious forces and the like, but different things happen.

No, they don't. Give me an experiment where I can make a measurement and get a result, and then do the same experiment in a different frame and get a different result.

Note: it must be the same experiment, and it must be something I can measure].

So there is some prefered reference frame in GR that you measure momentum in?

No. There are some coordinates in which it looks simpler written down under some circumstances.

When people talk about conservation laws I always think of it in terms of values and summations. In a non innertial frame I don't see why the summations of different times would have to be equal, because the reference frame is non innertial.

Think about it. If some quantity - call it "momentum" - is conserved in one reference frame, how can there not be an analogous quantity - call it "momentum" - which is conserved in the non-inertial frame? All I have to do is take the expression that says time derivative of momentum equals zero in the original frame and re-write it in the new frame, and obviously the equation is still valid (I'm just using different, human-chosen labels to write the same thing, like saying this in English or in Chinese).

So because of that fact, we need a set of laws that have the property that they predict that some quantity - call it momentum - is conserved no matter what coordinates we write the laws in. The necessary property is that the action from which the laws are derived be invariant under coordinate transformations.

 

[ponderingturtle]

No, they don't. Give me an experiment where I can make a measurement and get a result, and then do the same experiment in a different frame and get a different result.

Note: it must be the same experiment, and it must be something I can measure].

Any motion in a rotating frame. You are modifying your experiment to be frame independant in your mind. If you spinn your aparatus in the rotating frame to account for the rotation of the frame then you are not doing the same experiment.

Think about it. If some quantity - call it "momentum" - is conserved in one reference frame, how can there not be an analogous quantity - call it "momentum" - which is conserved in the non-inertial frame?

Possibly. But you are adding it time dependence, as it is not a conserved quantity but a changing quantity. In a simple rotating frame you get harmonic oscilations in the so called conserved values. So it is not truely conserved as the values keep changing.

All I have to do is take the expression that says time derivative of momentum equals zero in the original frame and re-write it in the new frame, and obviously the equation is still valid (I'm just using different, human-chosen labels to write the same thing, like saying this in English or in Chinese).

And it can is then not a conserved quantity in that reference frame. The value of the summation of all the momentums is not a constant.

You are saying that 1=2.

So because of that fact, we need a set of laws that have the property that they predict that some quantity - call it momentum - is conserved no matter what coordinates we write the laws in. The necessary property is that the action from which the laws are derived be invariant under coordinate transformations.

Just because you can work around to the right answer with enough work in non innertial frames doesn't make them equal.

 

[sol invictus]

Any motion in a rotating frame.

What do you mean, a motion "in a rotating frame"? Any motion can be described in a rotating frame or in a non-rotating frame.

Perhaps you don't know what "frame" means? It just means a set of coordinates, perhaps a set in which some object of interest is at fixed coordinate position.

You are modifying your experiment to be frame independant in your mind. If you spinn your aparatus in the rotating frame to account for the rotation of the frame then you are not doing the same experiment.

You've got it exactly backwards. Consider an apparatus floating in outer space that measures the length of a spring connecting a mass to a metal pole and displays the result on a digital readout. Let's say we make two identical copies of the apparatus, A and B. A is not rotating, B is rotating around the pole. A's screen will display a number smaller than B's.

We can describe this whole thing using non-rotating coordinates in which A is at rest. Or we can describe it in rotating coordinates in which B is at rest and A is rotating. If our laws of physics are going to corrrespond to reality, they had better reproduce the fact that A's screen will display a number smaller than B's in both sets of coordinates.

What Einstein showed is how to accomplish that.

Possibly. But you are adding it time dependence, as it is not a conserved quantity but a changing quantity. In a simple rotating frame you get harmonic oscilations in the so called conserved values. So it is not truely conserved as the values keep changing.

No, there is an exactly conserved quantity, where conserved means "doesn't change with time". That quantity is conserved in all frames, rotating or not. It's called "momentum".

Just because you can work around to the right answer with enough work in non innertial frames doesn't make them equal.

I don't know what you mean by "equal".

 

[Raze]

Thanks all for your clarification.


One more question:

Okay, if I stand on earth, I expect certain "physical laws" to govern how my body operates (for example, the electromagnetic force keeping me from walking through walls). Obviously if I walk through a fire, I'm gonna die. Etc.

Now, if I then step on a spaceship, and it accelerates, I would still expect those same certain "physical laws" to govern how my body operates (electromagnetic force keeping me from walking through walls, etc). Sure, I'll be jostled about, and in order to describe what is going on in my reference frame I may have to invoke "fictitious" forces, but I would expect that on the spaceship if I walk through a fire, I'm gonna die.

And any mathematical description of me being burned alive better be valid when I'm standing on earth or when I'm in an accelerating spaceship.
That's what I mean when I say the laws of physics. Of course, I am no expert, to say the least.

 

[sol invictus]

Okay, if I stand on earth, I expect certain "physical laws" to govern how my body operates (for example, the electromagnetic force keeping me from walking through walls). Obviously if I walk through a fire, I'm gonna die. Etc.

Now, if I then step on a spaceship, and it accelerates, I would still expect those same certain "physical laws" to govern how my body operates (electromagnetic force keeping me from walking through walls, etc). Sure, I'll be jostled about, and in order to describe what is going on in my reference frame I may have to invoke "fictitious" forces, but I would expect that on the spaceship if I walk through a fire, I'm gonna die.

And any mathematical description of me being burned alive better be valid when I'm standing on earth or when I'm in an accelerating spaceship.
That's what I mean when I say the laws of physics. Of course, I am no expert, to say the least.

Yeah, that's right, although the issue here is not so much what happens to you when you're accelerated - even Newton could answer that - it's how to construct theories that allow you to use non-inertial coordinate systems. In other words how to choose a coordinate system in which the rocket is at rest but still produce the correct laws to describe it.

 

[Raze]

Yeah, that's right, although the issue here is not so much what happens to you when you're accelerated - even Newton could answer that - it's how to construct theories that allow you to use non-inertial coordinate systems. In other words how to choose a coordinate system in which the rocket is at rest but still produce the correct laws to describe it.

Ah, so basically it's more or less about making every coordinate system equally valid? i.e. I'm on a rotating planet, but as far as I'm concerned I'm sitting here at rest in front of my computer?

 

[sol invictus]

Ah, so basically it's more or less about making every coordinate system equally valid?

Yes, it's how to construct the laws of physics in a way that applies in any reference frame. In a world in which spacetime is rigid and always flat that might be convenient, but it's perhaps not very important. But since gravity turns out to warp spacetime, it becomes crucially important.

i.e. I'm on a rotating planet, but as far as I'm concerned I'm sitting here at rest in front of my computer?

You're at rest in that rotating frame, yes. One thing that means is that you will measure different forces than if you were at rest in a non-rotating frame. So we need a set of laws that "know" about the frame in the right way (i.e. that change appropriately when we change frames).