Throughout this video, all the forces and velocities discussed will refer to how they would be seen by observers who stay on Earth, as they watch Victoria in her spaceship.
As with other relativistic effects, the phenomena discussed in this video actually occur at any speed, but they are not large enough to be noticeable at speeds far below the speed of light.
Isaac Newton originally formulated his second law of motion by stating that the rate at which the momentum of an object changes is equal to the force on the object.
This relationship is still true in Albert Einstein’s Theory of Relativity, but the definition of momentum now includes a variable we call "Gamma."
The momentum is equal to "Gamma" multiplied by the "Rest Mass" multiplied by "Velocity."
At speeds far below the speed of light, gamma is approximately equal to one, and the equation reduces to the familiar relationship stating that momentum is mass multiplied by velocity.
Gamma increases as velocity increases, and as the velocity approaches the speed of light, gamma approaches infinity.
Gamma multiplied by the "Rest Mass" is sometimes referred to as "Relativistic Mass."
Many people insist that the word "Mass" should always only refer to "Rest Mass."
But, when Einstein first presented his Special Theory of Relativity, he used the term "Relativistic Mass" and he also used the terms "Transverse Mass" and "Longitudinal Mass."
These two other terms refer to the fact that near the speed of light, a force perpendicular to the direction of motion has a very different effect than an identical force parallel to the direction of motion.
The letter "c" represents the speed of light.
Here, we see the relationship between the velocity and Gamma.
For the following analysis, let's assume that the "Rest Mass" of the ship and everything inside always stays constant, and ignore the fact that the rocket exhaust actually ejects mass out of the spaceship.
The momentum in the X direction is proportional to the area of this red rectangle.
If we apply a constant force, the momentum will change at a constant rate, and hence the area of the rectangle will change at a constant rate.
At speeds far below the speed of light, gamma is approximately equal to one, and the rectangle's area changes at a constant rate by having the velocity change at approximately a constant rate.
But at higher speeds, the area of the rectangle changes significantly both because the velocity is changing and because gamma is changing.
Since the area of a rectangle is its width multiplied by its height, a large width means that only a tiny additional change in the height is required to produce a large increase in the rectangle’s area.
And a large height means that only a tiny additional change in the width is required to produce a large increase in the rectangle's area.
So far, both the velocity and the force have been in the X direction.
Let's now instead briefly apply the same force in the Y direction, and consider the instant when this force is exactly perpendicular to the velocity.
When the force is exactly perpendicular to the velocity, the rate of change in "Gamma" is negligible.
The momentum in the Y direction is proportional to the area of this yellow rectangle.
Since the magnitude of the force is the same as before, the area of the yellow rectangle will change at the same rate as the red rectangle did previously.
This means that the Y component of velocity will change faster than the X component of velocity did previously.
This is to keep the rate of change of the rectangle's area the same, despite the fact that the rate of change in gamma is different in the two cases.
Therefore, a force perpendicular to the velocity produces a greater acceleration than does the same force applied parallel to the velocity.
With the force now again parallel to the velocity, gamma is significantly changing, and the velocity therefore changes more slowly.
Here, a constant force means that the area of the rectangle will decrease at a constant rate.
To keep the rectangle's area decreasing at a constant rate as the rectangle's height gets smaller, the rectangle's width decreases at faster rate.
We have the following two different equations for acceleration, depending on if the force is perpendicular or parallel to the velocity.
This is where the concept of transverse and longitudinal mass comes from.
On a historical note, this is not exactly the way that Einstein originally described transverse and longitudinal mass, because in our case, we are describing all the forces, velocities, and accelerations from the perspective of the observers who stayed on Earth.
And yes, this is Adam and Sarah that we saw at the beginning, who made their first appearance in the video, "Albert Einstein’s Theory of Relativity", describing all the other effects of travelling near the speed of light.
Victoria made her first appearance in the video "Physics: Laws of motion - Newton and Beyond", where you can see many more of her exciting adventures.
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