How to add an object's multiple relativistic speeds from x, y, and z?

[pervect]

Both of those are changing my question that I intentionally composed to avoid moving in any opposite directions:

"you reach 99% light speed in one direction, then without slowing, you start to accelerate into an either directly left or right direction (say the engine can rotate without turning the spacecraft) until you reach 99% into that direction as well. And again without slowing, you start to accelerate into an either directly up or down direction until you also reach 99% into that direction"

Emphasis mine.

Are we on the same page or are we missing details by speed reading at relativistic speeds which might happen to be length contracting our attention?

The exact calculation amount isn't vital, I merely wanted to confirm that relativistic effects from the additions of speed won't overshoot past c.

When you reach 99 percent of the speed of light in one direction (we'll call it the x direction), you'll have some momentum in the x direction $\gamma \, m \, .99 \, c$, where m is your mass.

If you start to accelerate in the y direction, if you maintain the same momentum in the x direction, your velocity in the x direction will decrease as you accelerate in the y direction, because the value of $\gamma$ will increase as you increase your velocity. Since your mass remains constant, this implies that the x component of the velocity must drop.

Here the "x,y,z" directions are relative to some fixed inertial frame of reference S.

If, instead, you accelerate in some manner to keep your x-velocity component relative to the fixed inertial frame S constant, you will never be able to exceed a velocity of $\sqrt(1 - .99^2)\,c$ in the y direction.

These are the two most likely interpretations of what you mean by "accelerate in the y direction". If you have some other interpretation, you'll have to state what it might be.

The general rules you need in special relativity are $F = dp/dt$, where p is momentum, and $p = \gamma m v$, where v is a vector, and $\gamma = 1 / \sqrt(1-|v|^2/c^2)$ where $|v|$ is a scalar that is the magnitude of the vector v, i.e. $|v| = \sqrt(v_x^2 + v_y^2 + v_z^2)$.

[add]Additionaly, there are various options for "t" you might use. I'm using the coordinate time of frame S. Other popular choices might be different coordinates, or the notion of "proper time", $\tau$, which is an interval along a worldline that a clock might measure.

 

[Demystifier]

TL;DR Summary: What are the relativistic effects if you go 99% light speed into the x axis, while you're going the same speed into the y axis (now two directions), and likewise the z axis. So three directions at 99% light speed each.

Obviously it'd add up to somewhere between 99% and 100% light speed. So how do you figure that out?

In the scenario, you reach 99% light speed in one direction, then without slowing, you start to accelerate into an either directly left or right direction (say the engine can rotate without turning the spacecraft) until you reach 99% into that direction as well. And again without slowing, you start to accelerate into an either directly up or down direction until you also reach 99% into that direction.

How do we add up the relativistic effects? (doubt they'd triple)

Are the 3 velocities all defined with respect to the same inertial frame? If so, the total velocity exceeds the speed of light, which of course is physically impossible.

The question only makes sense if the second velocity is defined with respect to inertial frame with respect to which the rocket moving with the first velocity is at rest, and similarly for the third velocity.

 

[syfry]

These are the two most likely interpretations of what you mean by "accelerate in the y direction". If you have some other interpretation, you'll have to state what it might be.

Now seeing the big flaw in my scenario. It's apparent by visualizing lower speeds on a globe, no math needed.

If the x we're traveling is south to north in a straight line along a longitude line, it's impossible to then accelerate into y along the equator or along any latitude line. We'd be moving diagonally, even if we aim due west or due east... straying from both of our lines.

In space the lines don't matter, but then we cannot also accelerate purely into x, y, or z without at the same time slowing in another axis.

There's overlap so we'd be partially accelerating into two axis which is impossible to avoid.

I tried accelerating into all 3 axis together at the same time, and we'd be traveling at some diagonal.

We're actually accelerating diagonally. So only one speed would ever apply.

That's why accelerating into another axis will slow the speed in a current axis.