# Relativistic mass?

#### rcgldr

Homework Helper

So for an object free falling from infinity to a black hole, even though it's velocity may become relativistic, it doesn't gain mass?
This one has me curious as well.

Does mass increase as velocity approaches light speed?

Does "weight" also increase?

Does the stregth of gravity from the object also increase?

If the mass and weight increase by the same ratio, then the rate of acceleration towards a black hole wouldn't be limited and would be ever increasing. What prevents such an object from reaching or exceeding light speed (as witnessed by an external observer)?

I tried to follow the wiki link, but it seems there's no simple (or single) explanation for mass in GR:

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

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#### skeptic2

1. By falling it is only converting the potential energy it already has to kinetic energy.
2. Even though it is being accelerated by gravity, to its own inertial reference frame it is at rest.
3. If it is gaining mass (energy) where is that mass coming from? Certainly not from the black hole.
4. If it gains mass as it falls, might it at some point have more mass than the black hole it is falling towards?

#### HallsofIvy

Homework Helper
We seem to keep saying this.

It is wrong to say simply that an object "increases its mass" or "shortens its length" or "times slows down" without specifying "as measured in what frame of reference".

The whole point of "relativity" is that such things are relative to the coordinate system in which the object is being observed.

#### rcgldr

Homework Helper
The whole point of "relativity" is that such things are relative to the coordinate system in which the object is being observed.
I thought I covered this by explaining that the frame of reference was from an external obsever. Say the obsever's position was relatively far away on a line that crossed the center of mass between the black hole and target object at the start of the initial condition.

Assume the observer is either far enough away or simple a frame of reference that the observer is not affected by the gravity from the black hole or target object.

Given these conditions, would the observer see a change in mass related to changes in velocities relative to the observer, as the two objects accelerated towards each other? Would the observer see either object appear to exceed the speed of light relative to the observer? If the objects had clocks on them, would the observer see the clocks rates change over time?

#### skeptic2

We seem to keep saying this.

It is wrong to say simply that an object "increases its mass" or "shortens its length" or "times slows down" without specifying "as measured in what frame of reference".
Yes, of course. Sometimes the frame of reference is apparent from the discussion and therefore not explicitly given. Many times the frame of reference is taken to be flat spacetime unless otherwise specified.

#### Sideways

"Relativistic mass" is a not a term that's used much any more, exactly because of the difficulties raised earlier on this thread. It's better to say that the mass is invariant, and that the energy and momentum is what varies from its classical prediction in the new reference frame.

#### Naty1

It's better to say that the mass is invariant, and that the energy and momentum is what varies from its classical prediction in the new reference frame.
Why?? Aren't all frame dependent?

#### Sideways

Why?? Aren't all frame dependent?
Yes, but it turns out that relativistic mass isn't too useful in calculations, and the concept just complicates things unnecessarily.

Momentum and energy are frame-dependent in classical physics even, remember; adding another frame-dependent quantity into the mix doesn't make things any easier. Better just to lump the relativistic variations in with energy, momentum, and leave the mass invariant.

#### Dale

Mentor
Sideways is correct. In modern usage mass is the norm of the four-momentum while energy is its timelike component and momentum is its spacelike components. Therefore mass is invariant, while energy and momentum are frame-dependent.

The deprecated concept of "relativistic mass" is the same as energy (timelike component of four-momentum).

#### jtbell

Mentor
Given these conditions, would the observer see a change in mass related to changes in velocities relative to the observer, as the two objects accelerated towards each other?
How are you imagining the observer to "see" or "measure" the mass of the object? In classical physics, there are various ways of doing this, which all give the same value for the mass. In relativity, these methods do not necessarily give the same value.

#### Naty1

If the mass and weight increase by the same ratio, then the rate of acceleration towards a black hole wouldn't be limited and would be ever increasing.
I thought a distant observer never sees a mass cross the event horizon; from a local free falling fram, the mass crosses the evnt horizon and disappears in finite time.

So I don't see that acceleration and velocity is unlimited.

#### rcgldr

Homework Helper
How are you imagining the observer to "see" or "measure" the mass of the object?
By observing any change made in the strength of the gravitational field near the target object (how it affects nearby objects). By observing the rate of change in acceleration due to gravity between target object and black hole as the target (and black hole) velocity increases with respect to the observer.

So I don't see that acceleration and velocity is unlimited.
I'm not stating a position here, I'm asking a question. Ignoring the relativistic mass issue, if there is a limit to the velocity (with respect to the distant observer) of an object just before it crosses the event horizon, what is the limit and why? Is there a limit to the rate of acceleration?

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#### Gokul43201

Staff Emeritus
Gold Member
Yes, there is a limit to the velocity - that limit is c.

Perhaps what you need to consider is that the momentum does not scale linearly with the velocity; it increases faster than the velocity does, by the additional factor $\gamma$. So even though the momentum (and energy) are increasing rapidly as the object approaches the black hole, the corresponding increases in the velocity get smaller and smaller as v approaches c (and gamma approaches infinity).

#### skeptic2

Yes, there is a limit to the velocity - that limit is c.

Perhaps what you need to consider is that the momentum does not scale linearly with the velocity; it increases faster than the velocity does, by the additional factor $\gamma$. So even though the momentum (and energy) are increasing rapidly as the object approaches the black hole, the corresponding increases in the velocity get smaller and smaller as v approaches c (and gamma approaches infinity).
In fact from the perspective of flat spacetime, not only do the increases in the velocity get smaller and smaller, the velocity appears to actually diminish as the object approaches the event horizon. Close to the EH space is increasingly contracted in the radial direction and what may seem like many kilometers to the object may be only a few meters for the observer, making the object appear to slow down.

#### atyy

In GR the speed of an object relative to a distant observer is not uniquely defined (and can be greater than the speed of light according to some definitions). At each point in its worldline, an object has a 4-velocity. However, due to the curvature of spacetime, the 4-velocities at vastly different points cannot be uniquely compared.

When considering a test particle falling into a black hole, the particle doesn't have any 4-acceleration, ie it moves on a geodesic. Every point in space is locally Minkowsian, so its local velocity never exceeds the local velocity of light.

In the case where you the infalling particle isn't a test particle, an interesting place to look might be the case of one black hole falling into another. In this paper by Baker et al, the final black hole "mass" is not the sum of the initial two black hole "masses", but I haven't understood the paper enough to know if this is significant or just numerical error. I mainly thought the video is fun to watch!

http://www.nasa.gov/vision/universe/starsgalaxies/gwave.html
http://arxiv.org/abs/0805.1428

Gold Member

#### feynmann

Slightly off topic but while looking for information on the web regarding mass and relativity, I found the following overview-

http://www.ph.rhul.ac.uk/course_materials/PH154/Relativistic mass and dynamics.pdf

Though it's entitled 'Relativistic Mass and Dynamics', it does make a good distinction between energy and mass.

a similar overview regarding Lorentz transformations-

http://www.ph.rhul.ac.uk/course_materials/PH154/Lorentz transformations.pdf
In the paper, it says that "Alternatively, we can say that the photon energy is equivalent to mass and is attracted by gravity like any other mass. Light is deflected by some 0.00048° as it grazes the sun."

Is it true that "photon energy is equivalent to mass and is attracted by gravity"?

#### stevebd1

Gold Member
Thanks for the response feynmann. Comparing photon energy to mass is a slightly slippery subject. The 0.00048° is based on

$$\delta=\frac{4Gm}{c^2r}$$

where δ is the starlight deflection in radians

which relates to Schwarzschild curved spacetime so I don't see the need to say that the photon energy is equivalent to mass and is attracted by gravity. I remember reading a description of gravitational lensing as being a consequence of gravitational time dilation; the side of the photon closest to the mass experiences less time which 'pulls' the photon off course; but again, this is treating the photon as a point mass rather than particle/wavelength that follows the curvature of spacetime.

Considering the http://en.wikipedia.org/wiki/Compton_wavelength" [Broken] $(\lambda=h/mc)$ which expresses the rest mass of a particle as a wavelength-

$$\tag{1}E = h f = \frac{h c}{\lambda} = m c^2$$

where h is Planck's constant, f is frequency and $\lambda$ is wavelength

you might (cautiously) consider the photon as being equivalent to mass but while it seems acceptable to consider sub-atomic particles as wavelengths in QM, I don't think it's acceptable to consider the (unbound) energy of a photon wavelength as a particle that could be affected by gravity. While I'm not saying the statement is wrong (the people who put the summary together are sure to be more educated in GR than I am), the statement does raise more questions than it answers.

Steve

(1)- http://en.wikipedia.org/wiki/Compton_wavelength

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http://arxiv.org/PS_cache/physics/pdf/0508/0508030v3.pdf" [Broken]

There's some interesting points made in this http://en.wikipedia.org/wiki/Talk:Photon/Photons_and_Mass_Debate2" [Broken] on wiki.

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