Magnitude of a velocity vector with special relativity

• Froskoy
In summary, the conversation discusses the use of velocity vectors in special relativity. The formula for magnitude of velocity in classical mechanics is v^2 = sqrt(v_x^2 + v_y^2 + v_z^2). In special relativity, the same formula applies for calculating the magnitude of velocity, regardless of the x, y, and z components. The rule for adding relative velocities in special relativity is not affected by separating a single velocity into its components.
Froskoy
Hi,

I'm trying to get my head around how velocity vectors work in special relativity. For example, in classical mechanics, the magnitude of the velocity would be given by:

$$v^2 = \sqrt{v_x^2 + v_y^2 + v_z^2}$$

where $v_x$, $v_y$ and $v_z$ are the $x$, $y$ and $z$ components of the velocity respectively.

What is the equivalent formula for use in special relativity. For example, if you knew the magnitude of the velocity, $v$ and the $x$ component of the velocity, what formula would you use to work out the $y$ component?

With very many thanks,

Froskoy.

Hi Froskoy!
Froskoy said:
… if you knew the magnitude of the velocity, $v$ and the $x$ component of the velocity, what formula would you use to work out the $y$ component?

same formula!

the special relativity rule for adding relative velocities has nothing to do with "de-componenting" a single velocity

What is the magnitude of a velocity vector with special relativity?

In special relativity, the magnitude of a velocity vector is the speed of an object in the absence of any external forces. It is a measure of how fast an object is moving in a particular direction, and is always measured relative to an observer's frame of reference.

How is the magnitude of a velocity vector calculated in special relativity?

The magnitude of a velocity vector in special relativity is calculated using the Lorentz transformation. This transformation takes into account the effects of time dilation and length contraction, which occur at high speeds, and allows for the accurate calculation of the magnitude of a velocity vector.

Does the magnitude of a velocity vector change with respect to different observers in special relativity?

Yes, the magnitude of a velocity vector is relative to the observer's frame of reference. This means that different observers moving at different speeds will measure different magnitudes for the same velocity vector.

How does the magnitude of a velocity vector contribute to the theory of special relativity?

The magnitude of a velocity vector is a fundamental concept in special relativity and is essential for understanding the theory. It helps to explain the observed effects of time dilation and length contraction, and is necessary for accurately predicting the behavior of objects moving at high speeds.

What are some real-life applications of the magnitude of a velocity vector in special relativity?

The magnitude of a velocity vector in special relativity has numerous applications in modern technology, such as in the design and operation of particle accelerators and in the development of space travel. It is also important in the field of astrophysics for understanding the behavior of objects moving at high speeds in the universe.

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