Electrons and Inertial Frames

[J_B_]

I do not think that you understand what a frame of reference is.

You may want to refer to the last paragraph of post #8 to help you with this sentence.

On an atomic scale the motion of an electron is not well defined so one cannot even begin to form a frame of reference from the electron's point of view.

OK. But after reflecting on some earlier posts, maybe it could be inferred.

I think it must. I don't know that you could tell directly, since electrons don't age. But relativistic effects provide little tweaks to chemistry. These are generally due to the increased mass of the electrons. And if you ask me, if your electron is getting fat from relativity, then it's clock is also slowing.

Posts such as this. If we could create a frame of reference for the electron, it probably wouldn't be inertial because the increased mass mentioned above implies relativistic effects. Yes?

So, if you could, please refresh my memory. How is the mass increase determined?

 

[J_B_]

Well, the electron is subject to a nonzero force, so I don't see how it could be.

D'oh. Yep, yep. Thanks, because that makes perfect sense.

I like Subduction Zone's answer possibly (possibly, mind you) better than my own.

I don't.

In the absence of everything and anything at all? Something like that.

Thanks.

Even then, the state is not uniquely determined.

Sure.

Most of the wavefunctions will look something like this:

Cool little graphic there.

 

[essentialsaltes]

it probably wouldn't be inertial because the increased mass mentioned above implies relativistic effects. Yes?

Non sequitur. Inertial frames can move as fast as you like (at constant velocity). Being relativistic does not mean it's not inertial.

 

[J_B_]

Non sequitur. Inertial frames can move as fast as you like (at constant velocity). Being relativistic does not mean it's not inertial.

Correction noted. Could you please refresh my memory regarding how we know electron mass increases?

 

[sjastro]

2) What is the probability of an electron's position in the absence of a nucleus, i.e. a free electron? Is it a straight line with constant velocity?

Electrons have a Compton wavelength of 2.43 x 10⁻¹² m which is another way of saying that even in a free state quantum effects dominate.
A physical explanation for this is to consider photons bouncing or scattering off objects.
For objects of large mass in the macro world nothing happens.
Photons striking the electron on the other hand imparts momentum to the electron and alters its position in space thereby causing an uncertainty in the electron's position and momentum.

 

[FrumiousBandersnatch]

It would be interesting to hear an argument about the legitimacy of reality, perhaps another thread another time?

'Reality is that which, when you stop believing in it, doesn't go away.' - Philip K. Dick

 

[Kaon]

Could you please refresh my memory regarding how we know electron mass increases?

Accelerating electrons gain mass.

 

[Subduction Zone]

You may want to refer to the last paragraph of post #8 to help you with this sentence.

I did. It contradicted this sentence in your OP:

"Is an electron orbiting a neutron in an inertial frame."

OK. But after reflecting on some earlier posts, maybe it could be inferred.

No. It really can't. At least not from an electron's "point of view". Perhaps we should go over the concept of an inertial frame of reference.

 

[sjastro]

Accelerating electrons gain mass.

It's not that straightforward.
For momentum to be conserved in all inertial frames (after the electrons are no longer being accelerated) the relativistic momentum is defined as:

The big question is whether the Lorentz factor 1/√(1-v²/c²) is applied to the mass m₀, or the velocity v.
Modern consensus is to treat the mass as an invariant like charge, as special relativity modifies our concept of time and space so that kinematic properties such as velocity are expected to change.

 

[J_B_]

Accelerating electrons gain mass.

It's not that straightforward.
For momentum to be conserved in all inertial frames (after the electrons are no longer being accelerated) the relativistic momentum is defined as:

The big question is whether the Lorentz factor 1/√(1-v²/c²) is applied to the mass m₀, or the velocity v.
Modern consensus is to treat the mass as an invariant like charge, as special relativity modifies our concept of time and space so that kinematic properties such as velocity are expected to change.

Right. No offense, but @Kaon didn't really answer my question. And this tells us how the momentum would change, but not how we know it (unless you presupposing relativity). I was asking what measurements reflect increased mass.

But I'm not sure it matters. We can move on.

 

[Subduction Zone]

Right. No offense, but @Kaon didn't really answer my question. And this tells us how the momentum would change, but not how we know it (unless you presupposing relativity). I was asking what measurements reflect increased mass.

But I'm not sure it matters. We can move on.

Changes in momentum, and therefore of mass, have been tested:

Tests of relativistic energy and momentum - Wikipedia

 

[J_B_]

Preamble: If you don't like preambles, skip this. Though I suspect for those who do, when you object to what I say next, I'll be replying with, "Please read the preamble in post #51."

As I've noted, all this stems from the book Spacetime and Geometry by Sean Carroll. I'm good with Chapter 1 now, so thanks everyone. It took some time to get through Chapter 1, not because of the math - that was easy enough - but because of the concepts. As you can tell, I would have walked away from Chapter 1 misapplying the math because of some misconceptions about the concepts. For me that's the most important part, as much as many scientists wish to avoid philosophy. It took extensive reading on the side and the discussion here to straighten me out ... though of course there's always a chance a misunderstanding is still lurking somewhere.

There were 2 concepts that threw me in Chapter 1. First, coordinate time vs. proper time. It wasn't so much relating time and space - I was already predisposed to thinking of space and the time as the same (though for theological rather than philosophical or scientific reasons). However, I'm not excited about the metaphysical underpinnings in Chapter 1. Let me be clear that I'm not challenging the math or the efficacy. I realize the explanatory power and I've already stated I don't have any physics credentials to speak of. However, had I been in Einstein's shoes, I would have gone a different direction. My outlook would have led me to a concept different from "spacetime" as it now stands. I'm not saying my approach would yield better results. I have no idea if it would work as well as spacetime - if it would work at all. I just never would have thought of it this way.

The second thing that took me awhile was grasping the implications of the hyperbolic nature of the Lorentz Transform. Carroll makes a comment that it preserves features similar to Euclidean orthogonality. Not that they were equivalent, but that they were similar - the difference being that the axes "scissor together". That was new for me, and it took me awhile to grasp it. But I'm there now.

So, my reaction to Chapter 1 was first, "Huh?" and then later "Cool." Not that I'm completely on board with the physical interpretations, but I understand them to the depth necessary that I can understand the consequences of viewing physical phenomena from that perspective.

But come Chapter 2, my reaction was ... Woa, woa, woa. Sorry, but no.

End Preamble
Begin Chapter 2: First Question

The first section of Chapter 2 attempts to convince the reader that the concept of spacetime logically follows from several concepts, the first being WEP (Weak Equivalency Principle).

Carroll discusses the equivalence of inertial forces and gravity. I was already familiar with discussions about force equivalencies - not inertia and gravity, but rather from D'Alembert's Principle. So, this wasn't new to me. Or maybe it is. Because Carroll argues the fact that since an inertia/gravity equivalency follows from WEP, that is a point in favor of spacetime. That this follows because m_i = m_g and a = [math]-del phi[/math]. Hmm. Seems to me that's an argument for gravity as a force, not the other way round.

Here's the question: Do you think WEP is an argument for spacetime? If so, why?

* Rats. My math codes didn't work. How do I type out equations here?

 

[sjastro]

But come Chapter 2, my reaction was ... Woa, woa, woa. Sorry, but no.

End Preamble
Begin Chapter 2: First Question

The first section of Chapter 2 attempts to convince the reader that the concept of spacetime logically follows from several concepts, the first being WEP (Weak Equivalency Principle).

Carroll discusses the equivalence of inertial forces and gravity. I was already familiar with discussions about force equivalencies - not inertia and gravity, but rather from D'Alembert's Principle. So, this wasn't new to me. Or maybe it is. Because Carroll argues the fact that since an inertia/gravity equivalency follows from WEP, that is a point in favor of spacetime. That this follows because m_i = m_g and a = [math]-del phi[/math]. Hmm. Seems to me that's an argument for gravity as a force, not the other way round.

Here's the question: Do you think WEP is an argument for spacetime? If so, why?

* Rats. My math codes didn't work. How do I type out equations here?

The WEP states a rocket accelerating at a=g is equivalent to a stationary rocket in a gravitational field g in the presence of a gravitating mass (a and g are in bold as they are vectors).

Consider an external inertial observer for both scenarios.
The inertial observer sees the laws of physics only applying to inertial frames.
In the first case where the rocket is accelerating at a=g, all the inertial observer needs to do is to apply a coordinate transformation to remove the accelerating components (the inertial or fictitious forces) in order for the rocket to be in an inertial frame.
In the second case for the rocket stationary in a gravitational field g things are not so simple; coordinate transformations won’t work and the only solution is to change the geometry of spacetime where a gravitating mass is present.

I assume Sean Carroll will explain this in his later chapters with the use of the components of a metric tensor which are solutions to the field equations.

Here is a simplified explanation.
…that all coordinate systems are created equal « Einstein-Online

 

[J_B_]

The WEP states a rocket accelerating at a=g is equivalent to a stationary rocket in a gravitational field g in the presence of a gravitating mass (a and g are in bold as they are vectors).

Consider an external inertial observer for both scenarios.
The inertial observer sees the laws of physics only applying to inertial frames.
In the first case where the rocket is accelerating at a=g, all the inertial observer needs to do is to apply a coordinate transformation to remove the accelerating components (the inertial or fictitious forces) in order for the rocket to be in an inertial frame.
In the second case for the rocket stationary in a gravitational field g things are not so simple; coordinate transformations won’t work and the only solution is to change the geometry of spacetime where a gravitating mass is present.

OK. I got more out of your explanation that Carroll's, though now I can see that he was saying something similar. But this still raises some questions. The first would be: Is this essentially an Ockham's Razor argument - that curved space is the simplest explanation? Or is it the only explanation?

 

[sjastro]

OK. I got more out of your explanation that Carroll's, though now I can see that he was saying something similar. But this still raises some questions. The first would be: Is this essentially an Ockham's Razor argument - that curved space is the simplest explanation? Or is it the only explanation?

This is a difficult question to answer.
Here is a list of the major theories of gravity devised.

Newton’s theory is by far the simplest and is still today the most extensively used theory in celestial mechanics; we use Newtonian gravity to calculate the trajectories of space probes.
On that criterion alone Newton’s theory is an example of Occam’s razor.
Newton’s theory however is incomplete and is a first order approximation of GR which is considerably more complicated.

The opposition theories to GR in the list considered “viable” are collectively known as tensor, vector and scalar gravity theories are based on curved spacetime (with the exception of Rastall) and satisfy the equivalence principle where at small scales curvature effects are not observed and the laws of physics as described by Special Relativity are valid in local flat (Lorentz) frames.

General Relativity is the preferred theory for gravity not because it is simpler or the only explanation, but it is heavily supported by observation.
In fact this table is probably out of date as the Tensor, Vector, Scalar theories classified as “valid” have been ruled out by the polarization modes of gravitational waves which were first discovered in 2016.

 

[lesliedellow]

I think it must. I don't know that you could tell directly, since electrons don't age. But relativistic effects provide little tweaks to chemistry. These are generally due to the increased mass of the electrons. And if you ask me, if your electron is getting fat from relativity, then it's clock is also slowing.

what could it mean to say an electron’s clock slows, when there are no processes are going on within it, and it doesn’t decay?

 

[essentialsaltes]

what could it mean to say an electron’s clock slows, when there are no processes are going on within it, and it doesn’t decay?

I don't know offhand, but probably people cleverer than I could develop some sort of test of it. But since relativistic mass and time dilation are both measured by the same 'gamma' factor, you can't have one without the other (or else relativity has to go back to the drawing board).

 

[sjastro]

what could it mean to say an electron’s clock slows, when there are no processes are going on within it, and it doesn’t decay?

The proper time τ for an electron is simply the time kept by a clock attached to the moving electron.
The time interval Δτ can be for example the travel time for an electron to move a distance Δx in space time.

Suppose the electron travels at a constant velocity v.
In the electron’s frame of reference the proper time is Δτ = Δx/v.
The coordinate time Δt = Δτ/√(1 – v²/c²) is the travel time measured by an observer say in a lab frame.

Since √(1 – v²/c²) < 1 for objects with mass, Δt > Δτ hence the travel time is dilated in the observer’s frame of reference.

 

[Radagast]

what could it mean to say an electron’s clock slows, when there are no processes are going on within it, and it doesn’t decay?

It certainly means something for muons, which do decay.

In electrons, relativistic effects, as I understand it, are responsible for fine structure in energy levels.