Question about full Lorentz transformation

[Sagittarius A-Star]

$\frac {\Delta x}{\Delta t} = \frac {u'+v}{1 + \frac {v}{c^2}u'}$

Are these equations correct?

Yes, that's correct. In Morin's book you can find this formula in the chapter "Longitudinal velocity addition", see pages XI-24 and XI-25 under the following link:
https://scholar.harvard.edu/files/david-morin/files/cmchap11.pdf
via:
https://scholar.harvard.edu/david-morin/classical-mechanics

Now you could plug into your last formula
$u' := c$ and then simplify again the right side of the equation.

Then you will see, how the speed of light transforms from the primed frame to the unprimed frame.

 

[Sagittarius A-Star]

Could the train be going so fast that moving to the front of the train could be impossible for a human because the amount of kinetic energy one would have to exert to move 1 meter toward the front of the train increases to the point where forward movement becomes impossible?

No. The passenger has the right to regard the train to be at rest. That's according to the principle of relativity. Then why should the very high velocity of the ground in $(-x')$-direction change anything for him/her?

 

[Chenkel]

No. The passenger has the right to regard the train to be at rest. That's according to the principle of relativity. Then why should the very high velocity of the ground in $(-x')$-direction change anything for him/her?

But doesn't the mass of an object increase the faster it's traveling with respect to the ground because the relativistic mass equation?

 

[Nugatory]

I thought $\Delta x, \Delta y, \Delta z$ is the location of an event in the unprimed frame and v is the velocity of the primed frame in the positive x direction of the unprimed axis.

x, y, z are the spatial coordinates, using the unprimed frame, of an event (and likewise x', y', z' are the spatial coordinate of an event using the primed frame). That is a location.

$\Delta x$, $\Delta y$, $\Delta z$ are the differences between the x, y, z coordinates of two events: for example, $\Delta x=x_1-x_2$ is the difference betweem the x coordinates of two events, one at $x_1,y_1,z_1,t_1$ and the other at $x_2,y_2,_z2,t_2$
That is not a location, it is the difference between two locations.

 

[Chenkel]

x, y, z are the spatial coordinates, using the unprimed frame, of an event (and likewise x', y', z' are the spatial coordinate of an event using the primed frame). That is a location.

$\Delta x$, $\Delta y$, $\Delta z$ are the differences between the x, y, z coordinates of two events: for example, $\Delta x=x_1-x_2$ is the difference betweem the x coordinates of two events, one at $x_1,y_1,z_1,t_1$ and the other at $x_2,y_2,_z2,t_2$
That is not a location, it is the difference between two locations.

So the Lorentz transformation also works on Delta x Delta t Delta z and Delta time between two events.

 

[jbriggs444]

But doesn't the mass of an object increase the faster it's traveling with respect to the ground because the relativistic mass equation?

Relativistic mass is not a very useful concept. Click here for an insight that explains why.

Special relativity is built upon an axiom that the laws of physics work the same regardless of what frame of reference you adopt. Relativistic mass (to the extent that it is a viable concept at all) rises from the math of special relativity. It does not contradict the postulates of the theory.

Regardless of how fast the train is moving relative to some inertial frame zipping to the rear at high speed, the train can be regarded as being at rest.

If you want to argue the opposite, you need to do the math. How much chemical energy is present in the muscles of the human who is about to run forward in the moving train? Please assess this using a frame of reference where the train is moving at 0.999 c. How much kinetic energy is present in a stationary human in the 0.999 c train? How much kinetic energy in a running human in the same train? Are those figures consistent with energy conservation?

Feel free to replace the human with ideal batteries, motors and toy cars if that will make things easier.

 

[PeterDonis]

So the Lorentz transformation also works on Delta x Delta t Delta z and Delta time between two events.

You should be able to figure that out for yourself by writing things out explicitly.

 

[Vanadium 50]

We are almost 100 messages in.

@Chenkel , I see you often respond almost instantaneously to messages. You need to think about them, not just react.

I also see you are posting new threads on this same subject - do you think that will be helpful? That your problem is that you aren't working on enough things at once?

Finally, I see a lot of good advice that you appear not to be taking.

I don't think you are on a path to success. Perhaps you should take on a different strategy, like rereading what people have already written more slowly and carefully and see if that helps.

 

[Sagittarius A-Star]

But doesn't the mass of an object increase the faster it's traveling with respect to the ground because the relativistic mass equation?

There were some discussions about relativistic mass in the past. Today most physicists agree to no longer use this terminology, but instead ${E \over c^2}$, which is the same.

Maybe, the following insights article, which was also linked by @jbriggs444, answers also your question, especially the last paragraph:
https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/

 

[Sagittarius A-Star]

So the Lorentz transformation also works on Delta x Delta t Delta z and Delta time between two events.

Yes. In addition to the good suggestion of @PeterDonis in posting #97 the following remark:

Also the LT without the Deltas can be interpreted as transforming coordinate-deltas, because of
$x = x-0$.

Then one of the two events is the common origin of the primed and unprimed frame: $t=x=y=z=t'=x'=y'=z'=0$.

Source:
http://www.scholarpedia.org/article...nematics#Galilean_and_Lorentz_transformations

For the LT without Deltas it is required to use the "standard configuration" for the two 4D-coordinate systems with a common origin $(0, 0, 0, 0)$.

This common origin is not required for a LT with explicit $\Delta$'s in it, because i.e. different offsets on the $x$-axis and $x'$-axis would not change coordinate-differences.

 

[PeroK]

We are almost 100 messages in.

Just to note that @Sagittarius A-Star must have contributed almost as much material to this thread as Morin has in the whole of Chapter 11!

This is why textbooks are written and why students should study them. Our job should then be to elucidate anything that isn't clear. Rather than construct a textbook ad hoc post by post.

 

[Chenkel]

Just to note that @Sagittarius A-Star must have contributed almost as much material to this thread as Morin has in the whole of Chapter 11!

This is why textbooks are written and why students should study them. Our job should then be to elucidate anything that isn't clear. Rather than construct a textbook ad hoc post by post.

I am indebted to all of the members of physics forums and I love the posts, equations and diagrams that @Sagittarius A-Star
contributed, I feel I'm understanding things much better and I will try to use text books more when I'm feeling curious instead of depending on the physics forums for understanding things.

Thank you all.