# Why electric field is scaled by gamma?

Two opposite charges are in a spaceship and are attracted by the electric field $$E_s$$ But for an observer on earth the Electric force is $$E_e=\gamma E_s$$ Normally the forces are scaled down by $$\gamma$$ in the earth frame and here also the total force is scaled down. But why the Electric component of force is scaled up? Is it because the Electric fields are now closer together because of length contraction?

But i think this is not the answer because if that was the case,then gravitational fields and hence Forces would have scaled up.But this is not the case in nature. Can you provide a derivation or something which explains how the Electric field is scaled up?

• Comments are not for extended discussion; this conversation has been moved to chat. Please do not use comments to discuss things unrelated to improving the post being commented on. May 3 at 15:16

In above Figure-01 an inertial system $$\:\mathrm S'\:$$ is translated with respect to the inertial system $$\:\mathrm S\:$$ with constant velocity
\begin{align} \boldsymbol{\upsilon} & \boldsymbol{=}\left(\upsilon_{1},\upsilon_{2},\upsilon_{3}\right) \tag{02a}\label{02a}\\ \upsilon & \boldsymbol{=}\Vert \boldsymbol{\upsilon} \Vert \boldsymbol{=} \sqrt{ \upsilon^2_{1}\boldsymbol{+}\upsilon^2_{2}\boldsymbol{+}\upsilon^2_{3}}\:\in \left(0,c\right) \tag{02b}\label{02b} \end{align}

The Lorentz transformation is \begin{align} \mathbf{x}^{\boldsymbol{\prime}} & \boldsymbol{=} \mathbf{x}\boldsymbol{+} \dfrac{\gamma^2}{c^2 \left(\gamma\boldsymbol{+}1\right)}\left(\boldsymbol{\upsilon}\boldsymbol{\cdot} \mathbf{x}\right)\boldsymbol{\upsilon}\boldsymbol{-}\dfrac{\gamma\boldsymbol{\upsilon}}{c}c\,t \tag{03a}\label{03a}\\ c\,t^{\boldsymbol{\prime}} & \boldsymbol{=} \gamma\left(c\,t\boldsymbol{-} \dfrac{\boldsymbol{\upsilon}\boldsymbol{\cdot} \mathbf{x}}{c}\right) \tag{03b}\label{03b}\\ \gamma & \boldsymbol{=} \left(1\boldsymbol{-}\dfrac{\upsilon^2}{c^2}\right)^{\boldsymbol{-}\frac12} \tag{03c}\label{03c} \end{align}

For the Lorentz transformation \eqref{03a}-\eqref{03b}, the vectors $$\:\mathbf{E}\:$$ and $$\:\mathbf{B}\:$$ of the electromagnetic field are transformed as follows \begin{align} \mathbf{E}' & \boldsymbol{=}\gamma \mathbf{E}\boldsymbol{-}\dfrac{\gamma^2}{c^2 \left(\gamma\boldsymbol{+}1\right)}\left(\mathbf{E}\boldsymbol{\cdot} \boldsymbol{\upsilon}\right)\boldsymbol{\upsilon}\,\boldsymbol{+}\,\gamma\left(\boldsymbol{\upsilon}\boldsymbol{\times}\mathbf{B}\right) \tag{04a}\label{04a}\\ \mathbf{B}' & \boldsymbol{=} \gamma \mathbf{B}\boldsymbol{-}\dfrac{\gamma^2}{c^2 \left(\gamma\boldsymbol{+}1\right)}\left(\mathbf{B}\boldsymbol{\cdot} \boldsymbol{\upsilon}\right)\boldsymbol{\upsilon}\boldsymbol{-}\!\dfrac{\gamma}{c^2}\left(\boldsymbol{\upsilon}\boldsymbol{\times}\mathbf{E}\right) \tag{04b}\label{04b} \end{align}

Nothing more, nothing less.

How the Lorentz force 3-vector or the Lorentz force 4-vector are transformed see my answer here Are magnetic fields just modified relativistic electric fields?.

Expressions of the kind $$''$$...scaled down by $$\gamma$$...$$''$$ or $$''$$...the Electric fields are now closer together because of length contraction...$$''$$ are misplaced.

• Peraphs MathJax does not support \boldsymbol. PS: missing the image. May 12 at 22:13
• @Sebastiano : It seems that the problem with `\boldsymbol' and the image is only yours (your browser ??? ). Nobody has reported anything like that. May 13 at 5:36
• Hi. now I see correct your answer: the image and the formulas. I use always the same browser Chrome. Yesterday I have seen only your two answers with the same problem. May 13 at 11:24
• @Sebastiano : Ok, browsing problems solved. May 13 at 11:27

Normally the forces are scaled down by γ in the earth frame and here also the total force is scaled down. But why the Electric component of force is scaled up?

The actual derivation is based on Lorentz transformation equations but one intuitive way to visualise this is to visulalise the electric field lines. In a charge at rest, the field lines are pointing outward uniformly in all directions.

But, in a charge that is moving , the field lines get "scrunched up" in the transverse direction to the field of motion, so that the electric field strength in that direction increases

• Thats what i initially imagined.But,if that was the cause,then gravitational force should also get stronger.But this doesn't happen.why is it so ? May 1 at 6:22
• Gravitation has nothing to do with electric field lines. Why would gravitational force behave like them ? May 1 at 6:30
• I mean we can represent electric field with electric field lines.Likewise gravitational force can be represented with gravitational field lines.so,if electric field get "scrunched up",then shouldn't gravitational field lines also behave like that? May 1 at 7:19
• I know Gravity is not a force.but,who knows if electricity is a force?Still we can represent both with field lines.. May 1 at 7:19
• Yes, but such field lines are simply our representations that help us predict what is going on. Without going into philosophy of science, these representations of field lines are not "real" and more importantly how the representations of electric field would behave are not necessarily identical with how the representations of gravitational field would behave, because these are 2 distinct and different fundamental forces with different "mechanism" May 1 at 7:28

Electric field E transforms this way:

$$E'=\gamma E$$

Gravity field G transforms this way:

$$G'=\gamma G$$

Force F, be it electric or gravitational, transforms this way:

$$F'= F / \gamma$$

• If G transforms as Ge = Gs*gamma , then how in the next line are you saying that gravitational force transforms Ge = Gs/gamma ? Those 2 statements are inconsistent with each other May 1 at 8:32
• @silverrahul What do you think my G means? I did say what it means, as I thought there might be some confusion otherwise. Read carefully : ) If you think that force can not go down when field goes up, then nature disagrees with you. : ) May 1 at 8:47
• Are you saying that Gravity field goes up , but force of gravity goes down ? How do you figure that ? If the force of gravity goes down, then the gravitational field at that point goes down, by definition. May 1 at 9:29
• @silverrahul Bob floats in space. A planet flies by with Joe standing on the planet. Bob says: "Planet's force on Joe is decreased, because of motion, planet's force me is increased, because of motion". May 1 at 10:39
• Transformation equations generally refer to the first case, where Joe and planet have same distance between them , only both are moving at higher velocity . Planet's force on Bob increasing, is not because of transformation. It is simply because distance between planet and Bob increases. May 1 at 10:46