# Why must the Lorentz Transformations Be Linear?

1. Sep 7, 2006

### emob2p

All derivations of the Lorentz Transformations I've seen assume a linear transformation between coordinates. Why must this be the case? Thanks.

2. Sep 7, 2006

### Hurkyl

Staff Emeritus
A transformation is linear if and only if it keeps the origin fixed, and it maps lines to lines.

3. Sep 7, 2006

### robphy

In particular, a Lorentz transformation must preserve the nature of timelike lines, which represent inertial particles.

4. Sep 7, 2006

### emob2p

So if the transformation were not linear, then an object would appear to be accelerating in one inertial frame but moving at a constant velocity in another. This would mean F=ma doesn't hold in both frames, a violation of a relativity assumption. Does that argument sound good to you guys?

5. Sep 7, 2006

### bernhard.rothenstein

linearity of the LET

I used to say to my students that the LET should be linear because to a pair of space-time coordinates in one inertial reference frame should correspond a single pair of space-time coordinates in an other one.

6. Sep 7, 2006

### Hurkyl

Staff Emeritus
LET? Oh, you mean Lorentz-Einstein transforms, not Lorentz Ether Theory.

That happens with any 1-1 transformation...

7. Sep 8, 2006

### bernhard.rothenstein

linearity

of couse! in any consistent theory.
sine ira et studio :

8. Sep 8, 2006

### Meir Achuz

Yes, the LT is linear so that if there is no accelartion in one LF, there will no acceleration in any other LF. But, don't talk about F=ma in SR.

9. Sep 8, 2006

### emob2p

why no F=ma in SR?

10. Sep 8, 2006

### emob2p

I guess it's more appropriate to say F=dp/dt.

11. Sep 8, 2006

### Meir Achuz

a and dp/dt are related by:
$\frac{d\bf p}{dt}=\frac{d}{dt}(m{\bf v}\gamma) = m\frac{d}{dt}\left[\frac{\bf v} {\sqrt{1-{\bf v}^2}}\right] =m\gamma^3[{\bf a}+{\bf v\times(v\times a)}].$

12. Sep 10, 2006

### leright

What? How do you figure that a frame can be accelerated to one inertial frame but not accelerated to another inertial frame. This makes no sense. F=ma holds in all inertial frames. However, as mentioned above, mass can vary. If mass did not vary then, due to the speed limit of c, a given force in an inertial frame @ .9999c would obviously yield a much smaller acceleration than that same force at .00001c, and if only acceleration changed due to a given force then the postulate of special relativity would be violated. But, at very high speeds the lack of acceleration is made up for in a large increase in mass, so regardless of speed, a given force yields the same momentum.

Last edited: Sep 10, 2006