Modern Special Relativity and Mass

  1. Some introductory texts and most "popular" literature has not kept up with modern conventions and mathematical techniques employed in the modern world of relativity. I see some of the confusions caused by this has influenced board members here as well as many other forums. So, I decided to write up a summery of how quantities in dynamics are formulated and defined in modern "special" relativistic physics. Hopefully this will serve to plant a nail in the coffin of obsolete conventions such as "relativistic mass" and "ict".
    Consider the rest frame of something we wish to write the equations of physics for. According to this frame the spatial components of its momentum will be zero. The amount of energy that it may have according to this frame is how we define mass m and the energy according to its rest frame is written [tex]E_{0}[/tex]. So
    [tex]m \equiv \frac{E_0}{c^2}[/tex]
    We define a four element vector constructed according to this frame by its time element [tex]P^{0}[/tex] given by
    [tex]P^{0} = \frac{E}{c}[/tex]
    and the three other components zero for this frame is what we will call the momentum. This results in the following.
    [tex]\left[P'^\mu\right] = \left[\begin{array}{cc}\frac{E_0}{c}\\0\\0\\0\end{array}\right][/tex]
    [tex]\left[P'^\mu\right] = \left[\begin{array}{cc}mc\\0\\0\\0\end{array}\right][/tex]
    The four vector momentum according to an arbitrary inertial frame is given by the Lorentz transform of this vector. So in general this results in
    [tex]\left[P^\mu\right] = \left[\begin{array}{cc}\gamma \frac{E_0}{c}\\\gamma\frac{E_0}{c}\frac{u^x}{c}\\\gamma\frac{E_0}{c}\frac{u^y}{c}\\\gamma\frac{E_0}{c}\frac{u^z}{c}\end{array}\right][/tex]
    [tex]\left[P^\mu\right] = \left[\begin{array}{cc}\gamma mc\\\gamma mu^x\\\gamma mu^y\\\gamma mu^z\end{array}\right][/tex]
    From this we can extrapolate a few things. The time component of the four vector momentum was what we called energy divided by c and the spatial components were what we call momentum. This then gives us the special relativistic expressions for momentum and energy according to an arbitrary inertial frame.
    [tex]E = \gamma mc^2[/tex]
    [tex]P^i = \gamma mu^i[/tex]
    Kinetic energy it the amount of energy we associate with motion only therefor we also arrive at the expression for the kinetic energy according to an arbitrary inertial frame
    [tex]KE = E - E_{0} = (\gamma - 1)mc^2[/tex]
    The four component coordinate velocity which is not a true vector is given by
    [tex]\left[u^\mu\right] = \left[\begin{array}{cc}c\\u^x\\u^y\\u^z\end{array}\right][/tex]
    The velocity four vector is defined by
    [tex]U^\mu = \frac{dx^\mu}{d\tau}[/tex]
    where [tex]\tau[/tex] is called proper time and can be thought of as time according to a hypothetical watch that rides along with the mass. The coordinate and four vector velocities can then be related through special relativistic time dilation.
    [tex]U^\mu = \gamma u^\mu[/tex]
    One can then refer back to the result for four vector momentum and arrive at
    [tex]P^\mu = mU^\mu[/tex]
    The relativistic force law is that four vector force is the proper time derivative of four vector momentum.
    [tex]F^\mu = \frac{dP^\mu}{d\tau}[/tex]
    Inserting the expression in term of four vector velocity results in
    [tex]F^\mu = m\frac{dU^\mu}{d\tau}[/tex]
    Four vector acceleration is defined as
    [tex]A^\mu = \frac{dU^\mu}{d\tau}[/tex]
    resulting in the following special relativistic force equation analog of Newton's second law.
    [tex]F^\mu = mA^\mu[/tex]
    Given this form of the dynamics equation for special relativity it becomes evident that a mass "changing with speed" is not the correct explanation for why a massive object can not be accelerated up to the speed of light. The mass m here does not change with speed. The correct explanation is that given this law of motion an arbitrary amount of ordinary force produces a diminishing coordinate acceleration as the coordinate velocity approaches c due to the time dilations involved in relating coordinate to four-vector expressions. These time dilations are in turn due to the Lorentzian structure of spacetime(the Lorentzian structure being why we defined the momentum four vector in terms of a Lorentz transform in the first place). So the reason really that nothing with mass can be accelerated up to c speeds in special relativity is ultimately that spacetime has a Lorentzian structure.
  2. jcsd
  3. Integral

    Integral 7,341
    Staff Emeritus
    Science Advisor
    Gold Member

    Thank you. While I have not completely comprehended your math, (I get tenser around tensors!) I believe your post contains answers to questions I have been asking myself.

    Edited to remove referrence to a deleted post
    Last edited by a moderator: Dec 16, 2003
  4. But that's just it: it's a convention. There's no correct way to go about it. There has been a long debate about this in the American Journal of Physics and Physics Today, and nothing has been settled, except that you can do physics either way; invariant mass or relativistic mass, it doesn't really matter which formulation you use, as long as you are consistent. Here's some references (pro and con) on the subject:

    Online refs.:

    Relativistic Mass
    What is mass?
    Rest mass or inertial mass?

    Printed refs.:

    Spacetime Physics - 2nd edition, Taylor and Wheeler, W H Freeman & Co. (April 1992)

    The Advantage of Teaching Relativity with Four-Vectors, Robert W. Brehme. Am. J. Phys. 36 (10), October 1968

    Does mass depend on velocity dad?, Carl Adler, Am. J. Phys. 55 (8), August 1987

    The concept of mass, Lev B. Okun, Physics Today, June 1989

    Letter to the Editor in Physics Today, Wolfgang Rindler, Physics Today, May 1990

    In defense of relativistic mass, T.R. Sandin, Am. J. Phys. 59 (11), November 1991

    Relativistic Generalizations of Mass, R.P. Bickerstaff AND G. Patsakos, Eur. J. Phys. 16 (1995), pg 63-66

    Basic Relativity, Richard A. Mould, Springer Verlag, (1996)

    Concepts of Mass in Contemporary Physics and Philosophy, Max Jammer, Princeton Univ. Press, (2000)

    Letters to the editors, Uspekhi: What is mass? R I Khrapko, Physics Uspekhi., 43 (12) (2000)

    Reply to R I Khrapko, Lev Okun, Physics Uspekhi., 43 (12) (2000)
  5. Re: Re: Modern Special Relativity and Mass

    I quite agree. It never ceases to amaze me how often this topic comes up. I mean, who really cares? It’s all just a matter of semantics. You don’t see people arguing over any other word in physics as much as you do the term mass. It all seems very silly to me.

    With regards to concept of mass, some of the same arguments apply to the concept of time as well. E.g. DW's post addresses what the term mass –should- mean. He/she is, of course, referring to proper mass (DW incorrectly implies that only introductory texts use the concept of relativistic mass. That is incorrect.). However it's just a word. It’s a word that different relativists use to mean different things in modern relativity. Only a student would be confused over such trivial things, not someone with a solid understanding of relativity. Such confusion can only be eliminated by study - not by renaming terms when a few students become confused. Perhaps DW is/was a student who is/was confused by relativistic mass.

    However this kind of multiple meaning of widely used terms happens throughout all of physics, not just relativity. E.g. when physicists use the term momentum it could refer to either mechanical momentum or generalized momentum. Sometimes physicists use the term Lagrangian when they really mean Lagrangian density. There's really no big deal here. It's all semantics. Nobody who knows physics solid is ever confused by things like this. Most would say "Big deal. Just say what you mean by it and don't worry about it."

    However if one chooses to work in 4-d spacetime geometry one uses geometric quantities like proper time and proper mass. The components of the 4-vectors might then be labeled time and mass. Jammer explains this in the text you listed as you may have seen. However unlike regular vectors Euclidean geometry, 4-vectors in spacetime geometry are of quite a different nature. In regular Euclidean geometry the components of vectors have the -same- physical meaning. However in spacetime geometry the components have a –different- physical meaning. One of my favorite sayings is that you can rotate a rod into a rod but you can't rotate a rod into a clock. Also, the geometric quantities don’t carry the same meaning in spacetime geometry as their names imply. For example; the magnitude of a massive particle’s four-velocity is the speed of light. The four-velocity of a massive particle can never be zero, even for a particle at rest. There is no four-velocity for the speed of light. However one can just as correctly work in the 3+1 view of relativity. In fact it’s sometimes very preferable and productive to do so.

    DW – the expression you gave for force is not a definition of force but a relationship between mass and force under certain conditions. In Newtonian mechanics F = mA is Euler’s expression for force. It only holds for particles of constant mass. Newton’s definition, the correct definition, is f = dp/dt.

    For the reason above it is best to distinguish between mass and proper mass just as its best to distinguish between time and proper time. Notice how DW uses a Greek symbol for proper time. To be consistent DW should have used a Greek symbol for proper mass. But nobody does that. I suppose its because it would be too confusing since the Greek letter for "m" is mu, which usually represents reduced mass.

    However it is incorrect to refer to the time component as energy. It is more appropriate to call it –mass-. The time component of (ct, r) is time. Therefore the time component of (Mc, p) should be called mass where M = m/(1-v^c/c^2)^1/2 (See Jammer). This convention is readily extendable to general relativity since the time component, P^0, of (mechanical) four-momentum is not energy as reading DW’s comments would suggest. P_0 is energy. The quantity M = m dt/d(tau) represents the inertia of a body, just as Einstein held.

    Notice how DW writes
    In his formulation one might get the impression that the speed of a particle is the magnitude of the four-velocity. However that magnitude is always equal to the speed of light. DW is mixing up the 3+1 view with the geometric view.

    DW also writes
    One is not right while the other is wrong. They are both equivalent ways of describing the same thing. However DW claims that the diminishing coordinate acceleration is due to time dilation. That is partially true since part of the diminishing coordinate acceleration is due to length contraction.

  6. Dear Arcon

    I would be very happy if you could help a novis with a simple ansver!

    I struggle to get a picture of the specetime.

    "Curvature tells matter how to move, and matter tells space how to curve".

    I will challenge you - not with a question but with an invitation - tell me the story that explains why my key drops to the floor when released from my hand! You are allowed to use entities as wraped space, equvalence principle and so forth but not any equations or math.

    I do want to understand gravity from the wraped spacetime perspective. No rubber sheets - I want the truth.... why does it fall down. What is ment by following a streight line in wraped spacetime? What causes the movemet? How does time get into the picture? I know these questions are kind of naive but I believe you can get my point; I know there is no Newtonian force but wraped spacetime but how do I get the real feeling for what happens when the key drops?

  7. Well, since the key is always in motion in space-time, the curvature tells the key where to move. I think the curvature doesn't make the key move, it just redirects it's motion. It follows the curvature.
  8. Re: Re: Re: Modern Special Relativity and Mass

    Any modern "relativity" text that does use it is either introductory or isn't a good text.

    The confusions would never arise if the improper terminology were never used in the first place. There would never have been a question posted here whether photon mass was infinite for example if the concept of mass changing with speed were never taught. And giving that poster the correct modern definitions seemed to satisfy him as to the answer.

    I was a student, but wasn't confused by it. You shouldn't speculate about my history.

    Actually there is a path parameterized four velocity of light.

    Any "particle" has constant mass. (outside of QM mass oscillation)

    No, because they are the same thing by definition.

    No because arguement by analogy is not a logical arguement. These are two different variables with two different definitions.

    You can say its inappropriate to call it either and then call it a dog if you want, but that doesn't change what it is.

    why? Jammer is not very good.

    (snipped some repetition)
    Its not "my" formulation. It is general relativity and if you are confused by it take your own advice and go study.

    I am not mixing anything up.

    No, length contraction has nothing to do with it. Describe for example the acceleration of a particle of no significant length. The relation between the coordinate acceleration and four-vector acceleration is determined only by the transformation of the time derivatives which are the only differences in their definitions.
    Last edited by a moderator: Dec 29, 2003
  9. Re: Re: Re: Re: Modern Special Relativity and Mass

    That is incorrect. The length of an object is not relevant to the question of diminishing coordinate acceleration, assuming a constant force that is. The relationship between force and acceleration is based on distance traveled and the time it takes to travel that distance and is reflected in the Lorentz factor 1/(1-v^c/c^2)^1/2. Since distances contract in the direction of motion and not perpendicular to the direction of motion, the resistance is therefore different in the direction of motion given the same acceleration in the rest frame in each direction.

    Consider two situations L and T. In situation L there is no force transverse to the direction of motion. In situation T there is no force in the direction of motion. Let the magnitude of the force, as measured in a frame in which the particle is moving, have the same magnitude in each case. Then the ratio of transverse acceleration to longitudinal acceleration equals the Lorentz factor. It follows from that the acceleration transverse to the direction of motion is greater that it is to the acceleration in the direction of motion by a factor of 1/(1-v^c/c^2)^1/2. The reason is due to Lorentz contraction of the distance traveled.

    Last edited by a moderator: Dec 29, 2003
  10. Re: Re: Re: Re: Re: Modern Special Relativity and Mass

    I was the one who said it wasn't relevent and it isn't for any angle. The only difference in SR dynamics from Newtonian dynamics is that the time derivatives in the SR definitions are proper time derivatives instead of coordinate time derivatives as the Lorentzian structure of spacetime yields time dilation. You might argue that Lorentz Vs Gallilean transformation is also indirectly relavent as in how [tex]dx^\mu[/tex] is related between frames, but as I show in the following equations the only explicit difference is in the time derivatives.

    Newton - coordinate velocity
    [tex]u^i = \frac{dx^i}{dt}[/tex]
    Modern Relativity - four vector velocity
    [tex]U^\lambda = \frac{dx^\lambda}{d\tau}[/tex]

    Nowton - coordinate acceleration
    [tex]a^i = \frac{du^i}{dt}[/tex]
    Modern Relativity - four vector acceleration
    [tex]A^\lambda = \frac{DU^\lambda}{d\tau}[/tex]

    Newtonian momentum
    [tex]p^i = mu^i[/tex]
    Modern Relativity - four vector momentum
    [tex]P^\lambda = mU^\lambda[/tex]

    Newtonian force
    [tex]f^i = \frac{dp^i}{dt}[/tex] - Your Newtonian expression
    [tex]f^i = ma^i[/tex]
    Modern Relativity - four vector force
    [tex]F^\lambda = \frac{DP^\lambda}{d\tau}[/tex]
    [tex]F^\lambda = mA^\lambda[/tex]

    Now the result I'm pointing out is that the mass m in the dynamics equation for relativity above does not change with speed. The reason the coordinate acceleration diminishes in SR and not in Newtonian mechanics obviously has nothing to do with a mass changing with speed. It has to do with the fact that the SR expressions differ from the Newtonian expressions by the time dilation in the proper time derivatives. Time dilates due to the Lorentzian structure of spacetime and so this structure, not a changing mass and certainly not length contraction is what is responsible for coordinate acceleration diminishing with speed.
    Last edited by a moderator: Dec 29, 2003
  11. I think having the acceleration decrease with increasing speed makes more sense asthetically as well. Large relative speeds change lengths, time and the addition of velocities. So it shouldn't be too surpising when relativity changes our understanding of meters, meters/s and seconds that meters/s^2 changes.

    Keeping the mass constant also avoids the "if mass travels fast enough can't it become a black hole?" mess.
  12. What mess? You mean the "mess" that students who cram the night before exams find themselves in? No physicist or conscientious physics student finds themselves in a "mess" because some books choose to present the "relativistic mass" formulation. I think it's valuable for the serious student to have two different ways of approaching this issue.
  13. I'm happy to see that someone realizes that fact. Due to the recent attempts to downplay the concept of relativistic mass Sandin wrote article in the American Journal of Physics (Rinlder wrote a letter in Physics today too) on the subject in hopes of warding off such attempt. The paper is

    In defense of relativistic mass, T.R. Sandin, Am. J. Phys. 59 (11), November 1991

    Rindler's article is found at
    (It has been placed there with the permission of the author)

    re - if mass travels fast enough can't it become a black hole?

    If one tries to use this arguement as an excuse to not teach relativistic mass then all one is doing is not uncovering a missunderstanding held by the person who asks such a question and gives the student the idea that the gravitational field of a moving particle is not a function of speed - which is incorrect. This too has been explained in the American Journal of Physics in the article

    Measuring the active gravitational mass of a moving object, D. W. Olson and R. C. Guarino, Am. J. Phys., 53, 661 (1985)

    If a student asks this 'black hole' question then all they are doing is revealing the misunderstanding that an object is a black hole becuase it has a large mass. That is a sufficient reason for an object to become a black hole since when an object reaches a certain mass it will collapse to a black hole. However it is not a neccesary condition for an object to become a black hole. Its quite possible for any object of any mass to become a black hole. All one has to do is crunch the matter to a small enough size. Hawking postulated the existance of mini black holes if I recall correctly.
  14. Gravitation in relativity is not at all the Newtonian [tex]\frac{GM}{r^2}[/tex] acceleration field of Newtonian gravitation where you are replacing the M with relativistic mass. This is bad physics and the fact that people propose this sort of thing is yet another example of what is wrong with teaching the very concept of "relativistic mass" outside of a mere historical footnote. Take a frame according to which a gravitational source is not in motion. Look at the motion of a test particle from the perspective of that frame. Now transform the results to a frame according to which the source is in motion. The source didn't somehow gain any matter making the test particle in any way heavier. The difference in the observed behavior of the test particle then is not in any way due to the motion of the source. The difference is mearly a matter of using a different frame to describe the motion of the test particle. The correct physics is (instead of writting an equation relating the Newtonian gravitational acceleration field to a mass source) is to write Einstein's field equations. These relate the correct source, the stress energy tensor (not relativistic mass) to second order derivatives of the metric. These differential equations are then solved for the metric at least in approximation and the results are used to arrive at the Riemann tensor. The Riemann tensor is the expression for Riemannian spacetime curvature and the modern relativistic concept of the gravitational field is the spacetime curvature field, not the Newtonian acceleration field. The reason modern relativity defines things this way is that it leaves the physics invariant to frame which is the very principle of relativity. The stress energy tensor, not relativistic mass, is the source. The Reimann tensor, not the Newtonian acceleration field, is the spacetime curvature field. The metric is the differential spacetime geometry. In the absence of real forces i.e. four vector forces the four vector law of motion for general relativity reduces to geodesic motion. Newton's law of gravitation only comes out of all this in the case of linearized weak field gravitation at low speeds. At higher speeds it isn't even a good approximation to replace mass with relativistic mass in Newtonian gravitation, because instead geodesic motion yields something that is even more analogous to electromagnetism than Newtonian gravitation.
  15. Good stuff here.

    But ... tell me. Doesn't a given body moving with a certain speed relative to me not actually exert greater "gravitational" force on me as it passes by at a certain distance, than if it were stationary relative to me at that same distance?

    I mean at all speeds, "relativistic" or not (hopefully to avoid possible simultaneity problems). If this is so, then can I not say the body exhibits larger gravitational mass when it is moving than when it is not? Furthermore, isn't gravitational mass equivalent to inertial mass?

    Alternatively or addtitionally, anybody care to reiterate exactly what the decomposition of the stress-energy tensor is into things more human beings can understand?


    If you're not confused, you don't understand the problem.

  16. Yes. Without question it most certainly does. However increase with velocity is neglegible at relativistic speeds.

    Mind you - the gravitational force is an inertial force. When I say that the gravitational force on a pariticle I'm speaking of the force as meaured by an observer who is in a frame in which the gravitational force has not be transformed away. For example: for simplicity of discussion consider the example of an infinite sheet of matter (i.e. the matter lies in a plane) with a uniform mass distribution. Suppose the matter is such that it generates what is appoximately a uniform gravitational field as observed by an observer who is in frame S where S is at rest with respect to the sheet in the frame in which the matter is not moving. Let g = the local acceleration due to gravity at z = 0 (sheet lies in z = 0 plane). g is proportional to the mass density rho.

    Now move relative to that sheet in the x-direction while remaining on the sheet at z = 0 - i.e. drive your car on the sheet real fast. Then in your car the local acceleration due to gravity will be increased by a factor of gamma^2 where gamma = 1/sqrt[1-(v/c)^2]. This is due to the fact that the mass density increased by a factor of gamma while the volume of the mass decreased by a factor of gamma leaving the mass density to increase by a factor of gamma^2. If a particle is weighed then the weight will be gamma^2 what it was in the rest frame.

    However due to the momentum of the matter there are other gravitational effects just as in EM where instead of only an increased electric field (only one factor of gamma there since charge does not increase with speed but density still does of course) but in the moving frame there is now a magnetic field. But if the particle is not moving in the new frame there will be no magnetic force on it.
    You betcha!
    Also - You betcha!
    To a certain extent you can think of it as you would charge and current. Think of the charge as relativistic mass and think of the current as momentum. (there is also "stress" in GR but you get the idea).
  17. Students will do bad physics whether or not we use the concept of relativistic mass. It's called not studying hard enough or last-night cramming. I just don't care about those "students". A student who understands time dilation, length contraction and the "twin paradox" will have no problems understanding relativistic mass.

    And not invariant or rest mass either. It seems to me that you have this idea that relativistic mass is an inherently confusing concept. I'm pretty dumb compared to the average person, and I understood it just fine. Indeed, I felt I understood things better when I was presented with more than one perspective (so even the pedagogical angle is covered).
  18. Bravo! Excellant obseration.
    And a student who does not understand time dilation will think that there are internal changes to the physical structure of the clock which causes the time dilation. A student who does not understand length contraction will think that there are internal changes to the physical structure of rods which causes the Lorentz-Fitzgerald contraction.
    The claim The stress energy tensor, not relativistic mass, is the source. is inaccurate.

    In the first place a mathematical quantity is not a physical entity which acts as a source of anything. The mathematical quantity is simply that: The mathematical quantity which describes the physical quantity.

    In the second place (relativistic) mass and energy are proportional and as such one can be replaced by the other by m = E/c2. In both special and general relativity (relativistic) mass and thus energy are both correctly and fully described by the stress-energy-momentum tensor Tuv. However one can also describe the same physics with the tensor Muv = Tuv/c2 and call that the "mass-momentum tensor".

    For purposes of illustration I've described such a tensor here

    In the third place it was Einstein himself who said, in his famous GR review paper on 1916, that
    The reason is similar to the EM case. For example: The mathematical entity which plays the role of source in EM is the 4-current J^u = (c*rho, j) where rho = charge density and j = current density. A charge at rest in one frame becomes charge and current in another frame. Same with relativistic mass. Rest mass in one frame becomes relativistic mass, momentum and stress in another frame.

    Charge is to EM as relativistic mass is to GR.

    I have a question for the entire group: Please explain why relativistic mass confuses you. If it doesn't confuse you then please tell us if it once did at one time and why you were confused at that time.

    Thank you
  19. Arcon:

    Thanks for your reply to my queries. I'm rusty as hell on most of this stuff, but thought I had a handle on it a few years ago. Your answers are pretty much as I expected them to be. ;-)

    In relation to your query in a later post:

    "I have a question for the entire group: Please explain why relativistic mass confuses you. If it doesn't confuse you then please tell us if it once did at one time and why you were confused at that time."

    Just see my 'signature'. ;-)
  20. selfAdjoint

    selfAdjoint 8,147
    Staff Emeritus
    Gold Member

    It's annoying to have to convert back and forth between the two systems. I have old relativity books that use relativistic mass, and it's a drag to convert their statements to the relativistic energy format to communicate with people trained on the modern system.

    The fact that the two systems are fully equivalent while saying different things just suggests to me that the math isn't very deeply coupled to the physics. Maybe we need to go to equivalence classes or something.
  21. That's the nature of modern relativity. What you use should relfect the problem at hand. Sometimes the geometric view of relativity is better to solve a problem and sometimes its better to use the 3+1 view of relavitity.

    This can also be explained rather easily by considering this from the stand point of the center of mass. See

    If the bullet is spinning at a constant rate then center of mass is moving at with uniform velocity. The momentum of the system can then be readily computated since the all the particles which make up the bullet are moving at constant speed and thus the calculation in the above link are valid.
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