# Gravitational and inertial mass

## Main Question or Discussion Point

In physics as a general discipline, there are 2 types of mass gravitational and inertial which have different definitions but experimentally they have turned to be extremely similar, 1 part in 10^12. Moreoever, general relativity predicts they are equivalent. They are all measured by an observer observing the masses accelerate in space.

However, in SR where all objects move with constant speed wrt each other there are two (different?) types of masses. The invariant mass and relativitic mass

M=m/sqrt(1-(v/c)^2), M=relativitic mass, m=invariant mass

How does the two masses in SR relate to the gravitational mass=inertial mass (lets assume they are equal for simplification)?
For one thing is it possible to measure gravitational mass=inertial mass in flat spacetime since acceleration of the masses must be needed but disallowed.

But from reading Wiki it seems that inertial mass = relativistic mass

http://en.wikipedia.org/wiki/Relativistic_mass
http://en.wikipedia.org/wiki/Mass_in_General_Relativity
http://en.wikipedia.org/wiki/Mass

Related Special and General Relativity News on Phys.org
In physics as a general discipline, there are 2 types of mass gravitational and inertial which have different definitions but experimentally they have turned to be extremely similar, 1 part in 10^12. Moreoever, general relativity predicts they are equivalent. They are all measured by an observer observing the masses accelerate in space.

However, in SR where all objects move with constant speed wrt each other there are two (different?) types of masses. The invariant mass and relativitic mass

M=m/sqrt(1-(v/c)^2), M=relativitic mass, m=invariant mass

How does the two masses in SR relate to the gravitational mass=inertial mass (lets assume they are equal for simplification)?
For one thing is it possible to measure gravitational mass=inertial mass in flat spacetime since acceleration of the masses must be needed but disallowed.

But from reading Wiki it seems that inertial mass = relativistic mass

http://en.wikipedia.org/wiki/Relativistic_mass
http://en.wikipedia.org/wiki/Mass_in_General_Relativity
http://en.wikipedia.org/wiki/Mass

If you examine the Eotvos class of experiments (the ones that verify the equivalence between gravitational mass and inertial mass), you would notice that the inertial mass that intervenes in these experiments is the rest mass (proper mass) and not any form of relativistic mass.

pervect
Staff Emeritus
Gravitational, inertial, invariant, and relativistic masses are all the same for an isolated system in a frame where the total momentum of the system is zero. Note that finding (or defining) the momentum of a system in GR is a bit tricky.

Therfore there isn't any need to "choose". Though personally I feel that relativistic mass is outdated and tends to be confusing, it is possible to use relativistic mass in a consistent and correct manner (it's just rather rare to see it actually done).

The relativistic mass of a system is not a property of the system alone - it is a property of the system and the observer. A lot of people that use relativistic mass seem to think (consciously or unconsicously) that there is some concept of "absolute velocity". This is of course, incorrect - there is no way to define the absolute velocity of an obserer. This is the reason that relativistic mass is not a property of a system alone - to specify the velocity of a system, one must specify some particular frame. This is not needed to specify the invarinat mass of a system, which is independent of the frame. (At least, the invariant mass of a system is indpendent of the frame for an isolated system - or for a system of zero volume, such as a point particle, regardless of whether or not the particle is isolated.)

A belief in absolute velocity and relativistic mass leads to frequent confusion on such issues as "If I move fast enough do I become a black hole" which we see periodically.

The gravitational field of a movng mass is an interesting question which deserves a topic in its own right- and has been previously discussed. The very short version is that it is very wrong to substitute some value of 'm' into a Newtonian formula and to expect to get correct results. Much like the electric field of a moving charge, the gravitational field of a moving mass is not even spherically symmetrical. To get into more details takes some time - in particular, the notion of "gravitational field" needs to be better defined before one can even start talking about the formulas for it.

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>> Much like the electric field of a moving charge, the gravitational field of a moving mass is not even spherically symmetrical.

What is moving or not depends on cooodinate system. Can always choose coordinates where the mass does not move and get the Schwarzschild solution.

pervect
Staff Emeritus
>> Much like the electric field of a moving charge, the gravitational field of a moving mass is not even spherically symmetrical.

What is moving or not depends on cooodinate system. Can always choose coordinates where the mass does not move and get the Schwarzschild solution.
Yep. What is of interest for "moving charges" is how the field transforms with a change of coordinates, though, rather than just the idea that one can chose different coordinates so that the charge is not moving (which is true, but not very helpful).

The first step in this process is defining what is meant by the field. For the electric field, we can compare the motion of a charged particle to an uncharged particle - this unambiguously defines the field at any point in space-time.

Doing this, we find transformation rules for a "boost" (the coordinate transformation which represents a change in velocity) such as illustrated in

http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_15.pdf
http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_14.pdf

Note that while the original field is spherically symmetric, the field transformed by a boost (a change in coordinates representing a change in velocity) is not.

For the gravitational field, though, there is no such thing as a gravitationally uncharged particle - all particles respond to gravitation. Thus measuring or even defining the value of the gravitational field "at a point" is tricker. The easiest solution is to measure the motion of a particle compared to a nearby particle - to measure not gravity, but tidal gravity.

Tidal gravity can be described by particular components of the Riemann tensor. Because it is a tensor, one can think of the Riemann as being defined "at a point". Tensors transform in a standard way - the Riemann, being a tensor, follows a standard set of transformation rules. I've worked out the details of what this means for tidal forces in

Note that it's rather technical - and also, mixed with some stuff that isn't particularly related.

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Gravitational, inertial, invariant, and relativistic masses are all the same for an isolated system in a frame where the total momentum of the system is zero. Note that finding (or defining) the momentum of a system in GR is a bit tricky.

Therfore there isn't any need to "choose". Though personally I feel that relativistic mass is outdated and tends to be confusing, it is possible to use relativistic mass in a consistent and correct manner (it's just rather rare to see it actually done).

The relativistic mass of a system is not a property of the system alone - it is a property of the system and the observer. A lot of people that use relativistic mass seem to think (consciously or unconsicously) that there is some concept of "absolute velocity". This is of course, incorrect - there is no way to define the absolute velocity of an obserer. This is the reason that relativistic mass is not a property of a system alone - to specify the velocity of a system, one must specify some particular frame. This is not needed to specify the invarinat mass of a system, which is independent of the frame. (At least, the invariant mass of a system is indpendent of the frame for an isolated system - or for a system of zero volume, such as a point particle, regardless of whether or not the particle is isolated.)

A belief in absolute velocity and relativistic mass leads to frequent confusion on such issues as "If I move fast enough do I become a black hole" which we see periodically.

The gravitational field of a movng mass is an interesting question which deserves a topic in its own right- and has been previously discussed. The very short version is that it is very wrong to substitute some value of 'm' into a Newtonian formula and to expect to get correct results. Much like the electric field of a moving charge, the gravitational field of a moving mass is not even spherically symmetrical. To get into more details takes some time - in particular, the notion of "gravitational field" needs to be better defined before one can even start talking about the formulas for it.
In a frame of a closed system where the total momentum is zero imply there was a time when everything was stationary. Hence at that time, the objects would have had masses that were inerital=gravitational=invariant mass.

But when the objects start moving, even though the total momentum is zero, the each individual moving object will gain mass which we call relativistic mass which will be heavier than the invariant mass. Does the gravitational and inertial mass change for those objects?

What about you a frame that is not the rest frame or center-of-momentum frame? What would be the Gravitational and inertial mass in those frames?

Here are two quotes from http://en.wikipedia.org/wiki/Relativistic_mass near the top.
"Einstein first deduced in 1905 that the mass (inertia) of bodies increases with their internal energy (energy content), but this mass too, is a kind of invariant mass (see section below on mass in systems)." Later it was stated that this was only to be done in COM frame which is what you have referred to. Would it mean gravitational=inertial mass also increase with internal energy content?

"Certain experiments will also observe an increased inertia for the object associated with the increase in relativistic mass."
This is saying increase in relativistic mass will increase the inertial mass=gravitational mass?

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What causes relativistic mass?Isnt it just increase in energy which gives the effect of energy?

what i believe is that mass is a form of energy.Mass occurs between a specific range of energy density but the range is not clearly defined.Even if energy density is slightly different characteristics of mass may occur.All energy forms differ due to energy density.And when an object attains relative velocity the energy density tends to increase which gives an effect of mass

I am correct.Please present ur views

jtbell
Mentor
In a frame of a closed system where the total momentum is zero imply there was a time when everything was stationary.
How do you arrive at this conclusion?

Hootenanny
Staff Emeritus
Gold Member
What causes relativistic mass?Isnt it just increase in energy which gives the effect of energy?
what i believe is that mass is a form of energy.Mass occurs between a specific range of energy density but the range is not clearly defined.Even if energy density is slightly different characteristics of mass may occur.All energy forms differ due to energy density.And when an object attains relative velocity the energy density tends to increase which gives an effect of mass

I am correct.Please present ur views
As I said in a previous thread (which was subsequently locked), I strongly dislike the use of relativistic mass, since relativistic mass is not clearly defined, although it can be used in certain applications if applied correctly. However, for fear of this thread being locked I will quote something Einstein said and comment no further on the matter;

"It is not good to introduce the concept of the mass M = m/(1-v2/c2)1/2 of a body for which no clear definition can be given. It is better to introduce no other mass than `the rest mass' m. Instead of introducing M, it is better to mention the expression for the momentum and energy of a body in motion."

As Zz suggested in the other thread. There is plenty information available if you search this forum.

what i believe is that mass is a form of energy.Mass occurs between a specific range of energy density but the range is not clearly defined.Even if energy density is slightly different characteristics of mass may occur.All energy forms differ due to energy density.And when an object attains relative velocity the energy density tends to increase which gives an effect of mass

I am correct.Please present ur views
What makes you think that?

How do you arrive at this conclusion?
Should I say there can be a time when everything is stationary instead?

Chris Hillman
A pedantic quibble

What is moving or not depends on cooodinate system.
Right, this is true even for Galilean kinematics.

As quantum123 says, we
can always choose coordinates where the mass does not move
(c.f. "comoving center of mass chart" in the discussion of the far field approximation in Stephani, General Relativity), but this next bit is not quite right (unless I missed some context):

and get the Schwarzschild solution.
Two basic problems here are that not all massive objects are isolated (admit a coordinate chart which is asympotically flat), and not all spacetimes containing an isolated object are stationary (admit a timelike Killing vector).

Now consider vacuum solutions which model a stationary asymptotically flat vacuum solution. From these select the axisymmetric solutions, which form the so-called asymptotically flat Ernst vacuum solutions. These represent axisymmetric and asymptotically flat stationary gravitational fields produced by a possibly spinning isolated massive object. From these select the solutions which model non-spinning objects; these form the
smaller family of Weyl vacuums. But even among the Weyl vacuums, most are not spherically symmetric. The static spherically symmetric vacuum solutions (this characterization does not depend upon using a particular chart!) are of course the one-parameter family of Schwarzschild vacuums.

The point is: even in Newtonian gravitation, not every isolated object produces a spherically symmetric gravitational field. Similarly for gtr, except that in gtr the gravitational field allows far more variety because in gtr, it matters whether or not the matter in the object is rotating or not.

pervect
Staff Emeritus
In a frame of a closed system where the total momentum is zero imply there was a time when everything was stationary. Hence at that time, the objects would have had masses that were inerital=gravitational=invariant mass.
Huh? Consider a spinning round wheel. The total momentum is zero, and it is a closed system, but it is not true that "at some time everything is stationary".

But when the objects start moving, even though the total momentum is zero, the each individual moving object will gain mass which we call relativistic mass which will be heavier than the invariant mass. Does the gravitational and inertial mass change for those objects?
Huh again? There seems to be some major communication problems here.

If you have an isolated system in special relativity, then the energy and momentum of that system will be constant and won't change with time.

Note that I've been talking about the mass of isolated systems in my responses. There are some subtle issues about the mass of non-isolated systems that I've been avoiding, because this thread is already confused enough without them and it would be inappropriate to introduce advanced material on the fine points when there is so much confusion about the basics.

Specifically what I said was:
Gravitational, inertial, invariant, and relativistic masses are all the same for an isolated system
I'll say this much since the topic of non-isolated systems has arisen: special relativity does offer some defintions of the mass of a non-isolated system, which must be applied carefully. General relativity however does not offer a general defintion for the mass of a non-isolated system.

In most cases, when one has a system interacting with the environment, it is possible to think of the system as being intially isolated, where it has some mass m_init, then allow it to interact over some time period, then isolate it again, where it then has a (generally different) mass m_final, so this isn't too restricting.

It's important to note that both the relativistic mass and the invariant mass of an isolated system will remain constant.

When the system is isolated, if one part of the system starts moving, there is an equal and opposite reaction per Newton's laws (or the appropriate relativistic generalziaton) such that the momentum of the system remains constant. The isolated system has to include the momentum and energy of all fields by which parts of the system interact with each other - it also has to be a closed system. There is thus never any change in the total energy or the total momentum of the system. If one part of the system starts moving, it causes an equal and opposite reaction on some other part of the system.

When a system is isolated, it also does not lose or gain energy.

I'd suggest that you considere re-reading the Wikipedia articles you quoted, or possibly a textbook such as Taylor & Wheeler's "Space-time Physics." There is a particularly good section in that particular book, a dialog on the use and abuse of the concepts of mass, which has many questions and answers. (The only thing that you might beware of is that the term "momenergy" the authors propose as a short hand for momentum/energy never really caught on).