Inertia and Momentum

[StringJunky]

How do they differ? They both seem to imply resistance to change in velocity; direction and/or speed.

Edited April 18, 2016 by StringJunky

[studiot]

An object with zero velocity has zero momentum, but it still has inertia.

 

Inertia is a measure of mass which is the factor in momentum that is classically independent of velocity.

Edited April 18, 2016 by studiot

[swansont]

Inertia is a somewhat nebulous concept. In some ways it mean mass, and in some ways it means momentum. You can sort of distill those two behaviors out of the concept of inertia.

[StringJunky]

Inertia is a somewhat nebulous concept. In some ways it mean mass, and in some ways it means momentum. You can sort of distill those two behaviors out of the concept of inertia.

Because velocity is a relation between objects, could one or the other concept be defined by their relative velocities. i.e if velocities are the same, each would measure zero momentum of the other but some different inertial value depending on their differences in mass? in short, are they frame-dependent concepts?

[swansont]

Because velocity is a relation between objects, could one or the other concept be defined by their relative velocities. i.e if velocities are the same, each would measure zero momentum of the other but some different inertial value depending on their differences in mass? in short, are they frame-dependent concepts?

 

 

Mass isn't, but velocity certainly is.

[StringJunky]

 

 

Mass isn't, but velocity certainly is.

You can measure the mass of an object if it is moving relative to yourself and it will be the same, as you measured it stationary? Hope that makes sense.

Edited April 18, 2016 by StringJunky

[swansont]

You can measure the mass of an object if it is moving relative to yourself and it will be the same, as you measured it stationary? Hope that makes sense.

 

 

Yes. In fact, many precise measurements are made when the mass is moving — a mass spectrometer measures the deflection of an ion of a known kinetic energy in a known magnetic field. Mass is an invariant even in relativity.

[StringJunky]

It's an absolute or constant parameter in this scenario?

[elfmotat]

It's an absolute or constant parameter in this scenario?

I'm not sure what you're asking. Mass remains constant under a change in reference frame, if that's what you mean.

[StringJunky]

I'm not sure what you're asking. Mass remains constant under a change in reference frame, if that's what you mean.

Indeed, . Thank you guys. Well, that's a little bit more understood.

 

Afterthought: Is energy, as measured from any frame, invariant as well?

Edited April 19, 2016 by StringJunky

[swansont]

Indeed, . Thank you guys. Well, that's a little bit more understood.

 

Afterthought: Is energy, as measured from any frame, invariant as well?

 

 

No. Kinetic energy is speed dependent, so energy in general is not invariant.

[StringJunky]

 

 

No. Kinetic energy is speed dependent, so energy in general is not invariant.

Right. I was just wondering if E+MC² contained all invariant properties, if mass and c are. The total energy would be conserved in all frames but manifested in different form, depending on the frame, would it not?

Edited April 19, 2016 by StringJunky

[swansont]

Right. I was just wondering if E+MC² contained all invariant properties, if mass and c are. The total energy would be conserved in all frames but manifested in different form, depending on the frame, would it not?

 

 

Within a frame energy is conserved. Between frames it is not.

 

m²c⁴ = E² - p²c² is invariant, while the individual terms on the right-hand side (E and p) are not.

[StringJunky]

 

 

Within a frame energy is conserved. Between frames it is not.

 

m²c⁴ = E² - p²c² is invariant, while the individual terms on the right-hand side (E and p) are not.

OK thanks.

[MigL]

So, just to be clear, both mass and energy are properties of a system and inexorably linked.

Yet one is frame dependent and the other is not.

 

Is this because of what we choose to define as 'mass' ?

 

( welcome back elfmotat )

Edited April 19, 2016 by MigL

[swansont]

So, just to be clear, both mass and energy are properties of a system and inexorably linked.

Yet one is frame dependent and the other is not.

 

Is this because of what we choose to define as 'mass' ?

 

( welcome back elfmotat )

 

 

Mass is one form of the energy. But you can define a mass to be frame dependent — that's the relativistic mass, which is a proxy for total energy, and thus redundant.

[Sensei]

Within a frame energy is conserved. Between frames it is not.

 

m²c⁴ = E² - p²c² is invariant, while the individual terms on the right-hand side (E and p) are not.

Take for example decay of unstable isotope in a box.

 

At the beginning of experiment atom has rest-mass m₀

thus energy E₀=m₀c²

(it's in the same frame of reference as box)

 

After decay one decay product has rest-mass m₁,

but relativistic mass is [math]m_1\gamma_1[/math] and energy is [math]E_1=m_1c^2\gamma_1[/math]

 

Second one has rest-mass m₂,

but relativistic mass is [math]m_2\gamma_2[/math] and energy is [math]E_2=m_2c^2\gamma_2[/math]

 

Total energy E₀, prior decay, is equal to E₁ and E₂ (plus what they gave away to medium) at any time after decay:

E₀=E₁+E₂

Initial mass is sum of relativistic masses of decay products:

[math]m_0=m_1\gamma_1+m_2\gamma_2[/math]

 

Decay energy is:

[math]D.E.=E_0-m_1c^2-m_2c^2[/math]

 

[math]\gamma_1[/math] and [math]\gamma_2[/math] are not constant. They are decreasing from their maximum at decay moment to 1.0 with time. While particle decelerate passing through medium in a box.

 

If you will start analyzing it from frame of reference of particle that decayed, box will be accelerated in direction of particle movement instead, and particle stationary.

So, just to be clear, both mass and energy are properties of a system and inexorably linked.

Yet one is frame dependent and the other is not.

 

Is this because of what we choose to define as 'mass' ?

 

When swansont says "mass" he means "rest-mass". Somebody else might interpret it as "relativistic-mass".

That's why I am always saying rest-mass or relativistic-mass to not confuse reader, which one I had in mind.

Edited April 19, 2016 by Sensei

[elfmotat]

So, just to be clear, both mass and energy are properties of a system and inexorably linked.

Yet one is frame dependent and the other is not.

 

Is this because of what we choose to define as 'mass' ?

 

Yes. There are a number of definitions of 'mass,' each with different properties, some of which are very rarely used, and some which aren't very useful. Just to name a few, in SR there's rest mass, relativistic mass, transverse and longitudinal mass. In GR things get even more complicated when trying to come up with definitions of non-local mass. As a result there are the Komar, ADM, and Bondi definitions, among others.

 

https://en.wikipedia.org/wiki/Mass_in_special_relativity

https://en.wikipedia.org/wiki/Mass_in_general_relativity

 

Usually when people talk about 'mass,' what they are referring to is rest mass, and usually the context will make clear to what one is referring.

 

Note that multiple definitions of a quantity is not particularly unusual. For example, in SR there's the coordinate acceleration, which is defined as [math]a^i := \frac{d^2 x^i}{dt^2}[/math]. This vector is not invariant under Lorentz boosts (which is not surprising because it is explicitly coordinate-dependent). There's also the four-acceleration: [math]a^\mu := \frac{d^2 x^\mu}{d \tau^2}[/math]. This vector is Lorentz-invariant, in that its magnitude does not change under boosts and corresponds to proper acceleration (i.e. the acceleration it actually 'feels', or that an accelerometer would actually read). In GR the four-acceleration is redefined as: [math]a^\mu := \frac{d x^\nu}{d \tau} \nabla_\nu \frac{d x^\mu}{d \tau} = \frac{d^2 x^\mu}{d \tau^2} + \Gamma^\mu_{\lambda \sigma} \frac{d x^\lambda}{d \tau} \frac{d x^\sigma}{d \tau}[/math]. This vector is invariant under all diffeomorphisms in that its magnitude corresponds to proper acceleration.

 

( welcome back elfmotat )

 

Thanks!

[Mordred]

Just a side note rest mass is often called invariant mass, while relativistic mass is now called inertial mass.

[MigL]

My first reaction was ...

Why are we using different 'conventions' to describe properties ?

And wondered why the alternate convention, i.e. the concept of 'rest energy' or 'invariant energy', not used in physics.

 

But as you guys explained, and once you think about it, any measured mass is simply the sum of rest or invariant mass and kinetic energy.

We could, I suppose, simply say that 'mass-energy' has two components, we call the invariant component mass, and the variable one K energy.

[swansont]

My first reaction was ...

Why are we using different 'conventions' to describe properties ?

And wondered why the alternate convention, i.e. the concept of 'rest energy' or 'invariant energy', not used in physics.

 

But as you guys explained, and once you think about it, any measured mass is simply the sum of rest or invariant mass and kinetic energy.

We could, I suppose, simply say that 'mass-energy' has two components, we call the invariant component mass, and the variable one K energy.

 

 

 

We do use rest/invariant mass in physics. Relativistic mass is a zombie intruder. It eats brains and it spreads, never dying. But because much of standard physics is based on the other definition, if you mix them up it causes great confusion.

 

I've noticed that a number of pop-sci articles talk about mass increasing, (e.g. articles about why gold is gold colored, owing to relativistic corrections to orbitals) but the original article never mentions mass. Just energy. So at least part of this is the attempt to explain things in a way that a lay audience can understand. An effort of convenience and/or laziness (because relativity isn't intuitive for most people). But it often ends up doing more harm than good.