# Relativity and Inertia - An Exploration

• Urmi Roy
What is the relationship there?In summary, Relativity-I think! says that inertia is related to the energy an object has, and that it increases with the velocity of a body.f

#### Urmi Roy

Relativity-I think!

Hi,
I'm new here,so I'm kinda nervous.

Anyway,I have a big question(big to me atleast!) about inertia.

I've been pondering what exactly this thing inertia is-why does a body have inertia just because it has a mass? What capability does "matter" in an object have to try and prevent external forces from acting on it?

I thought that Einstein's E=mcsquared might help-saying that mass has a certain amount of energy may mean that it has energy to oppose external forces--I'm not really sure.

On the moon where the gravitational forces are a lot weaker,its easier to lift an object,does it mean that it has less inertia. If it does, inertia is supposed to depend on mass only,so gravitational forces aren't supposed to affect it!

Inertia is the resistance of an object to a change in motion. Why do objects have inertia? I don't think anyone really knows that. Inertia of an object doesn't change on the moon though. You will still need to apply the same force to give it a certain acceleration as you had to on earth.

If your car (mass 1000 Kg) is coasting on level ground at 10 meters per sec, it has momentum or inertia of 10,000 Kg-m/sec. It will continue to coast (no friction) at 10 mps until it interacts with something (another car?) that absorbs some or all of the inertia. It does not have inertia just because it has mass. It has inertia because it resists changing its velocity. It has nothing to do with gravity and weight, altough a falling object of mass m has velocity v and therefore has inertia of mv. Also, a spinning wheel on your car has rotational inertia, because it has a moment of inertia I (approximately 1/2 mR2) and a rotational angular velocity w so the rotational inertia = Iw.

Bob S said:
If your car (mass 1000 Kg) is coasting on level ground at 10 meters per sec, it has momentum or inertia of 10,000 Kg-m/sec. It will continue to coast

Absorbing inertia seems to be a strange concept. It would imply that it is easier to change the car's speed after the collision than it was before the collision (not counting the car being torn apart and reduced in mass).

Consider two particles with a property called "electric charge". This property results in an electric force between them, causing them to accelerate toward each other: "inertial mass" is defined as property that says how the electric force is related to the acceleration.

Now consider two particles with a property called "gravitational charge". This property results in an gravitational force between them, causing them to accelerate toward from each other: "inertial mass" is defined as a property that says how the gravitational force is related to the acceleration.

So "gravitational charge" is just a property like "electric charge" that produces forces between particles - it is a distinct concept from "inertial mass" which is defined as a property that tells us how forces and accelerations are related.

Experimentally, "gravitational charge" is always proportional to "inertial mass" even though they are different concepts. Because of this constant relationship between "gravitational charge" and "inertial mass", "gravitational charge" is usually called "gravitational mass".

The moon has less gravitational mass/inertial mass than the earth. Even though the mass of a small particle is the same on the Earth and the moon, the gravitational force on the small particle is less on the moon than on the Earth because the mass of the moon is less than that of the earth.

Hi,
I've been pondering what exactly this thing inertia is-why does a body have inertia just because it has a mass? What capability does "matter" in an object have to try and prevent external forces from acting on it?
Actually,if we define momentum as a form of inertia, then massless photons qualify. If a photon has energy hv, then its momentum is hv/c. A properly designed radiometer operating in a vacuum can demonstrate the force from sunlight (about 0.1 watts/cm2) striking a vane (about 1 x 10-4 dynes). So maybe having mass is not a requirement for having inertia.

Hi,

However no-one really said anything bout the relativity concept.

I think that Bob s's remark about car absorbing inertia kind of agrees with this concept of E=mcsquared. It seems the car 'absorbs' inertia because it has velocity.We also know that inertia increases with the velocity of a body,in a way that means the greater energy,the greater inertia.So inertia is a form of energy stored up on an object-the energy which thus helps it to prevent any change in its state of motion
Again,I come back to a slight confusion-- as we know,all motion is relativistic. Hence looking at it in that angle, a body at rest w.r.t to me hasis supposed to have inertia because of energy stored inside it--but since it is at rest,it doesn't have energy(neglecting gravitational effects,since it dosn't have any effect on inertia)! How do I explain that?

Last edited:

I vaguely remember reading somewhere that a container of hot gas has more inertia than the same container with the same amount of cold gas. So that would imply that energy has inertia. Which would explain why mass has inertia, but then begs the obvious question about why energy has inertia.

Again,I come back to a slight confusion-- as we know,all motion is relativistic. Hence looking at it in that angle, a body at rest w.r.t to me hasis supposed to have inertia because of energy stored inside it--but since it is at rest,it doesn't have energy(neglecting gravitational effects,since it dosn't have any effect on inertia)! How do I explain that?
All motion is relative, but rarely relativistic, meaning it satisfies Newton's equations but not necessarily special relativity equations. If I were driving along a deserted desert road on a moonless night, and I suddenly saw your car come up and hit me. I would tell my insurance company that I saw your car hit mine. You would tell your insurance company my car hit yours. So which car had more inertia? But wait. If you were driving a pickup, and I a VW bug, which car had more inertia? In this case, for Newtonian mechanics, inertia means resistance to change in velocity (I think), and not specifically resistance to change in momentum. So inertia has to refer to mass. My college mechanics book (Becker, Theoretical Mechanics) states:
"Inertia is the property of a body by means of which the body resists a change in its motion. The term is a qualitative one, and is conventionally employed in a qualitative sense. The quantitative measure of inertia is mass..."

If your car (mass 1000 Kg) is coasting on level ground at 10 meters per sec, it has momentum or inertia of 10,000 Kg-m/sec.
What you're describing there is momentum, not inertia - they are different concepts. Admittedly inertia isn't defined quantitatively the way momentum is (unless you consider mass = inertia), but I'm pretty sure it's not something that jumps around from object to object, that can be gained or lost in a simple collision.

While I think it is ok to talk about the momentum of a photon (planck constant/wavelength) I am not convinced that you can talk sensibly about the inertia of a photon.

The reason for this is that (according to wikipedia) "inertia is the resistance of an object to a change in its state of motion".

On creation, a photon goes from not existing to existing and traveling at c and after that resists all changes to its state of motion (it's complicated to think that the path of a photon can be bent by gravity but the photon itself does not change its state of motion, because it is space itself which is bent - but that's the way it is).

So it would seem that, apart from the moment of creation, the inertia of a photon is infinite (or "capped" as in you can't get more resistance to change than "impossible to change").

Or is there another way to look at it?

I note that inertia seems to be qualitive rather than quantative - there is no actual equation for how much inertia something has.

cheers,

neopolitan

In this case, for Newtonian mechanics, inertia means resistance to change in velocity (I think), and not specifically resistance to change in momentum. So inertia has to refer to mass. My college mechanics book (Becker, Theoretical Mechanics) states:
"Inertia is the property of a body by means of which the body resists a change in its motion. The term is a qualitative one, and is conventionally employed in a qualitative sense. The quantitative measure of inertia is mass..."

I got a little confused with this part-please could someone clarify this for me.

Again,the last 3 threads each seem to be going in different directions-I request you to please sum it up for me. Thankyou so much for your help.I really appreciate it.

I got a little confused with this part-please could someone clarify this for me.

Again,the last 3 threads each seem to be going in different directions-I request you to please sum it up for me. Thankyou so much for your help.I really appreciate it.

You might want to ignore my post, I was just pondering about how inertia and momentum are not the same thing (because you can sensibly talk about the momentum of a photon but maybe not the inertia of a photon).

That might say more about the nature of a photon than the nature of inertia and momentum.

cheers,

neopolitan

To clarify a point, momentum is not inertia.

diazona said:
What you're describing there is momentum, not inertia - they are different concepts. Admittedly inertia isn't defined quantitatively the way momentum is (unless you consider mass = inertia), but I'm pretty sure it's not something that jumps around from object to object, that can be gained or lost in a simple collision.

I agree to this. In one of the posts it was stated that an object can transfer some or all of its inertia. If we combine this with the definition of inertia, that is the object's resistance to change in motion.This would imply that if an object has transferred all of its inertia it does no longer resist a change in motion. This just doesn't happen in nature. Imagine you're playing a game of pool and you strike one ball in such a way that after it hits another ball it stops. It would have transferred all of its inertia after which the slightest touch would cause it to accelerate indefinitely. Yet I am pretty confident that unless the ball's mass has been reduced a force F would cause the same acceleration to the ball regardless whether I strike it before or after the collision.

please give me a sum up all your statements. It seems that it is difficult to answer my original question (of what exactly inertia is and how mass enables an object to resist eternal forces),without taking momentum into consideration.but I'm still messed up.

I've been pondering what exactly this thing inertia is-why does a body have inertia just because it has a mass? What capability does "matter" in an object have to try and prevent external forces from acting on it?

<snip>

On the moon where the gravitational forces are a lot weaker,its easier to lift an object,does it mean that it has less inertia. If it does, inertia is supposed to depend on mass only,so gravitational forces aren't supposed to affect it!

In physics, inertia is "the tendency of a body to maintain its state of rest or uniform motion unless acted upon by an external force".

In common parlance, inertia is conflated with:
• mass (more strictly "inertial mass" which is defined as "the resistance of the matter to acceleration or deceleration, as given by the factor m in Newton's 2nd law F = ma"), and
• momentum (strictly "the product of a body's mass and its velocity" but more loosely "an impelling force")

Mass is certainly something that has inertia, but as DaleSpam pointed out, that is not the whole story, since a container of hot gas has more inertia than a container containing the same amount of cold gas. So the heat itself may have inertia - I say this because the behaviour of hot gas is largely kinetic in nature - I'd more interested to hear whether a hot solid has more inertia than a cold solid.

If heat conveys inertia, then we can say that energy is the deciding factor, not just mass.

Anyway, to better understand, I think it might be worthwhile to do a thought experiment in which there was no inertia. If nothing "resisted" forces on them, everything would be spinning around the universe at maximum speed (because a photon hitting the Earth would be enough to send the Earth off at light speed). That sort of shows that inertia is not something odd, but something completely necessary.

Also, I would have thought that inertia derives directly from the laws of thermodynamics - the energy of a system remains constant, so if you have a small object hit and merge with a large object then the energy ends up distributed across the new system comprising of both merged, the inertia of the now slightly larger large object just reflects the fact that much more energy than exists in the system is required to give it the same velocity that the small object had before the collision.

cheers,

neopolitan

PS About the moon, you're not just talking about the inertia of the object, but also the fact that there is a lesser force already acting on the object you are trying to lift. To lift an object on the Earth you have to overcome not only the inertia of the object but also the force of gravity acting on it - on the Earth that is 9.8 Newtons per kg and on the moon that is 1.6 Newtons per kg. So to give your 10kg object 2m/s2 acceleration on the Earth, you need to apply (9.8 + 2) * 10 = 118 Newtons. On the moon, that will be (1.6 + 2)* 10 = 36 Newtons. You could, in a rough sort of way, say that the inertia you are overcoming is represented by the 2 * 10 = 20 Newtons - a figure which is, as you say, unaffected by gravity. A problem that would experience on the moon that we don't have so much on the Earth is stopping the thing you are lifting from continuing its upwards motion.

Last edited:

In broad terms you can think of inertia as a property of an object that measures a certain cause-effect relationship. In the case of mechanical inertia, it is a measure of the effect that a change in the mechanical energy of an object has upon its motion, in particular upon its velocity (eg. the bigger the mass of the object, the more mechanical energy it must absorb for the same linear velocity increase). Or think of thermal inertia (or heat capacity), which is a measure of the amount of thermal energy a body must absorb to undergo a certain temperature increase. Mass, moment of inertia (with respect to an axis), and heat capacity are all measures of types of inertia, and as others have pointed out, are inherent properties of a given object.

The bigger the respective inertia, the less senstive the object is to changing its state, be it its state of motion, or temperature, or whatever, to variations in the amount of energy it possesses.

Sorry but I don't know about special relativity :> (that is my pursed beak-face:P)

Mass is certainly something that has inertia, but as DaleSpam pointed out, that is not the whole story, since a container of hot gas has more inertia than a container containing the same amount of cold gas. So the heat itself may have inertia - I say this because the behaviour of hot gas is largely kinetic in nature - I'd more interested to hear whether a hot solid has more inertia than a cold solid.

Um, heat is a mechanism of transfer of energy, and temperature is a measure of internal energy (okay, microscopically you can say that all forms of energy a body possesses boil down to the kinetic energy of its constitutive molecules, but macroscopically we quantify random motion that doesn't contribute to advection as internal energy) . . . anyhow . . . heat does not "have" inertia. As far as I know only material objects can have inertia. A body has more or less energy, which is manisfested as kinetic energy (speed, be it linear or angular), or temperature. A hot body has more internal energy than a cold body of equal mass. Inertia measures the relation between the increment in energy content that the body possesses to the external manifestation of that increment in energy (linear speed, angular speed, temperature..)

Last edited:

please give me a sum up all your statements. It seems that it is difficult to answer my original question (of what exactly inertia is and how mass enables an object to resist eternal forces),without taking momentum into consideration.but I'm still messed up.

Mass is a measure of only one kind of inertia.
Note that:

$$K_t = \frac{1}{2} m v^2$$
and

$$K_r = \frac{1}{2} I w^2$$

where Kt is the translational kinetic energy of a body, and Kr the rotational kinetic energy, m is the mass, I the moment of inertia (with respect to the axis of rotation considered), v the linear velocity and w the angular velocity (with respect to the aforementioned axis). m and I are two measures of inertia, one for linear motion, the other for angular motion. Note that for an equal translational kinetic energy increase, a body with more mass will have a smaller linear velocity increase. Likewise, for an equal rotational kinetic energy increase, the body with a greater moment of inertia will undergo a smaller angular velocity increase. The concept of momentum does not enter here. Inertia simply gives us a measure of the relationship between how much of a certain kind of energy a body can store and the respective manifested change in its observable state (in this case different types of motion).

As to your statement, mass does not enable an object to resist external forces - mass enables an object to resist a change in its linear motion (and angular motion, since I is related to mass as well as geometry), whereas external forces are simply the agents that cause the change in motion (the providers of the energy). Mass quantifies the effect that the external force has upon the change in the body's motion.

Last edited:
Hot things are more massive

Sorry but I don't know about special relativity :> (that is my pursed beak-face:P)

heat does not "have" inertia. As far as I know only material objects can have inertia.
Actually, bizarre though it might seem, in relativity, heat does have inertia. If you heat an object up, its mass increases (slightly): the heat energy gets accounted for as extra mass, via E = mc2, and you need to push it that little bit harder to accelerate.

Actually, bizarre though it might seem, in relativity, heat does have inertia. If you heat an object up, its mass increases (slightly): the heat energy gets accounted for as extra mass, via E = mc2, and you need to push it that little bit harder to accelerate.

See...this is what I mean!

I appreciate BobbyBear's clarifications about the concept of inertia,but I think that the above stated quotation deals with what was actually bothering me.

It seems that wherever there is energy, there is inertia,both of which can 'become each other' using Einstein's equation.

So,in my frame of reference,for a body at rest, it appears to me to have a mass,which is a measure of inertia in (linear motion),so there must be someone else in relative (uniform)motion with me who actually sees this 'mass' as 'energy'...and in one particular frame there must be no mass at all,but only energy!

Again,with reference to this, I might just say,since there is such interchangeability between 'energy' and 'mass', mass has this unique ability to resist change in state of (linear)motion (of the object possessing it) because it has 'energy' inside it ,almost like a 'hidden power'.

I could also say that the molecules in the object posses this 'hidden power' so whoever wants to change the object's state of motion,will be resisted by them.

Actually, bizarre though it might seem, in relativity, heat does have inertia. If you heat an object up, its mass increases (slightly): the heat energy gets accounted for as extra mass, via E = mc2, and you need to push it that little bit harder to accelerate.

I'm not sure if I'm interpreting E = mc2 correctly, but as far as I understand, it states the equivalence of mass and energy . . . so doesn't that mean that you can't consider having both things at the same time? It tells you that you can look at a certain mass as an equivalent amount of energy, or a certain amount of energy as an equivalent amount of mass, but you can't look at a certain mass as having both mass and the equivalent amount of energy simultaneously, if you know what I mean. That is, in your example of heating an object up, you either look at the energy provided to the object as an equivalent amount of mass added to the original mass of the object, or as energy. So when you say you heat an object up, you either view the energy communicated to the object as thermal energy (an increase in temperature of the object: E=m cp (T2-T1) ), so that you now have a hotter object but with the original mass; or as an increase in mass of the object but with the object having the same initial temperature (that is, you view the heat energy provided as an equivalent amount of mass). You can't say that you've both increased its mass and temperature!? (please note that whenever I say 'mass' I am referring to what I think is known as 'rest-mass').

Hmm, so okay, that little point cleared I see what you mean: if you always consider the energy of an object as an additional mass rather than as energy possesed by the body (kinetic, thermal . . .) , then the more energy it has, the more inertia it has, which is indeed extremely bizarre (to me:P).

Also please note that I've tried to explain what inertia means and how it's quantified from a purely non-relativistic view-point, so I don't even know if what I've said in previous posts holds when you bring in relativistic concepts. Sorry for any confusion that may have caused.

I'm not sure if I'm interpreting E = mc2 correctly, but as far as I understand, it states the equivalence of mass and energy . . . so doesn't that mean that you can't consider having both things at the same time?
No, just the opposite.
It tells you that you can look at a certain mass as an equivalent amount of energy, or a certain amount of energy as an equivalent amount of mass, but you can't look at a certain mass as having both mass and the equivalent amount of energy simultaneously, if you know what I mean.
Sure you can.

No, just the opposite.

Sure you can.

O:

Well, me thinks me cannot hope to understand relativity, at least just yet :P
Still, is there any book you can recommend for lay people?

Please would anyone comment on my last post (post number 23 in this thread)?

I would love for any misconception I may be having to be cleared up,please.

So,in my frame of reference,for a body at rest, it appears to me to have a mass,which is a measure of inertia in (linear motion),so there must be someone else in relative (uniform)motion with me who actually sees this 'mass' as 'energy'...and in one particular frame there must be no mass at all,but only energy!
I would not agree with this. The "m" in E = mc2 refers to the "rest" mass, which is an invariant--the same for all observers. What does change with the frame of the observer is the kinetic energy of the body. The faster the body is seen to move, the greater its kinetic energy.

...
So,in my frame of reference,for a body at rest, it appears to me to have a mass,which is a measure of inertia in (linear motion),so there must be someone else in relative (uniform)motion with me who actually sees this 'mass' as 'energy'...and in one particular frame there must be no mass at all,but only energy!
...
Just to add to what Dr. Al just said, even you will observe the body at rest to have some energy; that's exactly what E=mc^2 tells you - the "rest energy" of a body. Once it's moving, its total energy is the sum of this term plus the velocity-dependent kinetic energy.

This is sort of an aside, in response to the reference in the OP to mass on the Moon:

Reportedly one of the most difficult adjustments the Apollo astronauts had to make was to take into account that their inertia was the same on the Moon as on Earth, even though their weight was reduced. That meant that they could jump up much more easily and felt much lighter, but when they tried to change direction as they hopped along, their mass resisted the change just as much as on Earth. I think I read years ago that some of them fell down the first time they tried to "turn right" - their intuition told them that they'd be just as "light" horizontally as they were vertically, which wasn't quite the case.

So,in my frame of reference,for a body at rest, it appears to me to have a mass,which is a measure of inertia in (linear motion),so there must be someone else in relative (uniform)motion with me who actually sees this 'mass' as 'energy'...and in one particular frame there must be no mass at all,but only energy!
Maybe this will help. One of the most important concepts in SR is the four-momentum. The timelike component of the four momentum is proportional to the total energy or the relativistic mass. The spacelike components are the momentum, and the norm (magnitude) of the four-momentum is proportional to the rest mass.

Now, like all four-vectors the four-momentum Lorentz transforms like time and space do, and the norm is invariant. So this implies the following for a massive particle:
1) The rest mass is invariant (all frames agree)
2) The relativistic mass is minimized in the rest frame and is strictly greater in other frames
3) There is no frame where either mass goes to 0

Hi,
I'm new here,so I'm kinda nervous.

Anyway,I have a big question(big to me atleast!) about inertia.

I've been pondering what exactly this thing inertia is-why does a body have inertia just because it has a mass? What capability does "matter" in an object have to try and prevent external forces from acting on it?

I thought that Einstein's E=mcsquared might help-saying that mass has a certain amount of energy may mean that it has energy to oppose external forces--I'm not really sure.

On the moon where the gravitational forces are a lot weaker,its easier to lift an object,does it mean that it has less inertia. If it does, inertia is supposed to depend on mass only,so gravitational forces aren't supposed to affect it!