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Then, it was pointed out by Thomson and Searle that this electromagnetic mass also increases with velocity. This was further elaborated by Hendrik Lorentz (1899, 1904) in the framework of Lorentz ether theory. He defined mass as the ratio of force to acceleration, not as the ratio of momentum to velocity, so he needed to distinguish between the mass ${\displaystyle m_{\text{L}}=\gamma ^{3}m}$ parallel to the direction of motion and the mass ${\displaystyle m_{\text{T}}=\gamma m}$ perpendicular to the direction of motion (where ${\displaystyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}$ is the Lorentz factor, v is the relative velocity between the aether and the object, and c is the speed of light). Only when the force is perpendicular to the velocity, Lorentz's mass is equal to what is now called "relativistic mass". Max Abraham (1902) called ${\displaystyle m_{\text{L}}}$ longitudinal mass and ${\displaystyle m_{\text{T}}}$ transverse mass (although Abraham used more complicated expressions than Lorentz's relativistic ones). So, according to Lorentz's theory no body can reach the speed of light because the mass becomes infinitely large at this velocity.

Albert Einstein also initially used the concepts of longitudinal and transverse mass in his 1905 electrodynamics paper (equivalent to those of Lorentz, but with a different ${\displaystyle m_{\text{T}}}$ by an unfortunate force definition, which was later corrected), and in another paper in 1906. However, he later abandoned velocity dependent mass concepts (see quote at the end of next section).

The precise relativistic expression (which is equivalent to Lorentz's) relating force and acceleration for a particle with non-zero rest mass ${\displaystyle m}$ moving in the x direction with velocity v and associated Lorentz factor ${\displaystyle \gamma }$ is

\begin{aligned}f_{\text{x}}&=m\gamma ^{3}a_{\text{x}}&=m_{\text{L}}a_{\text{x}},\\f_{\text{y}}&=m\gamma a_{\text{y}}&=m_{\text{T}}a_{\text{y}},\\f_{\text{z}}&=m\gamma a_{\text{z}}&=m_{\text{T}}a_{\text{z}}.\end{aligned}

Source: https://en.wikipedia.org/wiki/Mass_in_special_relativity#Transverse_and_longitudinal_mass

The question would require to look at it in historical context to some extent. I understand that the terminology and concepts discussed below are old fashioned but I'm still interested and curious.

Question:

We can only focus on 'x' and 'y' directions. Longitudinal mass is greater than the transverse mass. Suppose an electron, or any other mass body, is travelling only in the x direction at constant velocity of 0.4c. According to the given formula, to accelerate it in x direction, force required is $f_{x}=m_{L}a_{x}$. Assume that the initial velocity in the y direction is zero, to accelerate it in y direction the force required could be calculated as follows $f_{y}=m_{T}a_{y}$. It would be easier to accelerate it in the y direction. The electron has been accelerated in the y direction and it's now travelling in the y direction with same velocity of 0.4c. Note that now the electron is travelling in both directions, x and y, with same velocity of 0.4c. Does the mass or electron still have different inertia in different directions?

I'm under the impression that it's the velocity in a certain direction which defines inertia in that direction for a certain mass. For example, if the mass is needed to accelerate more in y direction so that its velocity in y direction becomes 0.5c then now one need to use $f_{y}=m_{L}a_{y}$ instead since any velocity over 0.4c in y direction is larger compared to the constant velocity of 0.4c in x direction. I hope you get my point and guide me where I'm having it wrong. Thank you.

Edit #1 (5:15 AM, Friday, October 2, 2020, UTC)

Let me rephrase the question differently.

Let's focus on only 'x' and 'y' directions. Longitudinal mass is greater than the transverse mass. Suppose an electron, or any other mass body, is travelling only in the x direction at constant velocity of 0.4c. According to the given formula, to accelerate it in x direction, force required is $f_{x}=m_{L}a_{x}$.

Let's assume that the initial velocity in the y direction is zero. To accelerate it in y direction the force required could be calculated as follows $f_{y}=m_{T}a_{y}$.

Why the required transverse force $f_{y}$ would be less than longitudinal force $f_{x}$? In other words, why the mass seems heavier in the x direction than in the y direction?

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    $\begingroup$ Your previous question on this topic was migrated to HSM so why are you reposting to this site? A search will reveal that relativistic mass is widely considered by members here to have been an unfortunate historical mistake in how to interpret Special Relativity. $\endgroup$
    – G. Smith
    Commented Sep 19, 2020 at 0:56
  • $\begingroup$ @G.Smith The reason being that, in my opinion, the other question was more about the concept but this question is more on numeric side. I have always thought that HSM is good for discussing concepts in their historical context. $\endgroup$
    – PG1995
    Commented Sep 19, 2020 at 0:59

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The electron has been accelerated in the y direction and it's now travelling in the y direction with same velocity of 0.4c. Note that now the electron is travelling in both directions, x and y, with same velocity of 0.4c. Does the mass or electron still have different inertia in different directions?

The electron must have the same inertia in x and y directions, because the situation is symmetric in such way that there can be no difference between x and y directions.

Inertia in x-direction increased when we increased the y-speed, because we increased the amount of energy that is moving at speed 0.4c in the x-direction.

Also it may be worth mentioning that it was necessary to apply a force in the x-direction in order to keep the x-speed constant.

If x-speed is kept constant, then x-inertia is proportional to total energy. (By x-inertia I mean inertia in the direction of the x-axis, ignore this if you are only interested about inertia transverse to the motion and inertia in the direction of the motion)

Notice that in our case the direction of motion of electron changes, so direction of 'transverse' changes too. At the beginning the transverse direction is the y-direction, at the end it's not.

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  • $\begingroup$ Thank you. "Inertia in x-direction increased when we increased the y-speed, because we increased the amount of energy that is moving at speed 0.4c in the x-direction." - Originally there was no motion in y-direction, the mass or electron was already moving constant speed of 0.4c in x direction, but still the force needed to accelerate over 0.4c in x direction was $f_{x}=m_{L}a_{x}$. "Also it may be worth mentioning that it was necessary to apply a force in the x-direction in order to keep the x-speed constant." - Not sure what you mean here. Aren't x and y directions of Cont'd $\endgroup$
    – PG1995
    Commented Sep 19, 2020 at 23:12
  • $\begingroup$ motion supposed to independent of each other? Assuming no air resistance or friction, constant velocity in x direction should remain unaffected by motion in y direction. $\endgroup$
    – PG1995
    Commented Sep 19, 2020 at 23:13
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    $\begingroup$ @PG1995 If x-speed is c, then y-speed can only be 0. If we increase y-speed, then x-speed decreases 'automatically'. By that I mean that X-motion becomes time-dilated when Y-motion increases. That's how motions are dependent of each other. In our case the direction of motion of electron changes, so direction of 'transverse' changes too. At the beginning the transverse direction is the y-direction, at the end it's not. $\endgroup$
    – stuffu
    Commented Sep 20, 2020 at 2:09
  • $\begingroup$ Thank you. Sorry for the late follow up but I'm still confused. I have marked up your reply and comment to appreciate your attempt to help me. I have also edited the question to make it easier to understand. $\endgroup$
    – PG1995
    Commented Oct 2, 2020 at 5:57
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    $\begingroup$ @PG1995 A fat guy accelerates a one gram pebble from 0.9c to 0.91c, becoming a skinny guy while doing so. Something pushed 40 kg of fat from 0c to 0.91c. Or 40 kg of energy, if that is better. Anyway, the guy felt the recoil of 40 kg of stuff being accelerated from 0 to 0.91c. (100% of the fat was converted to energy) $\endgroup$
    – stuffu
    Commented Oct 2, 2020 at 8:17

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