# Is the 4-velocity always a space-tipe 4-vector?

1. May 11, 2009

### martyf

Is the 4-velocity always a space-tipe 4-vector?

2. May 11, 2009

### dx

Re: 4-velocity

No, 4-velocity is always time-like, unless you're talking about tachyons.

3. May 11, 2009

### martyf

Re: 4-velocity

But |4-velocity|^2 = -1 . So why is it time-lke?

4. May 11, 2009

### martyf

Re: 4-velocity

How can I show that a 4-velocity is always time-like?

5. May 11, 2009

### dx

Re: 4-velocity

4-velocities of ordinary particles are timelike because they travel slower than light.

6. May 11, 2009

### martyf

Re: 4-velocity

Ok, so if I have a given 4-velocity how can I show in terms of mathematics that it is time-like ?

7. May 11, 2009

### dx

Re: 4-velocity

Show that it's components satisfy t² - x² - y² - z² = 1.

8. May 11, 2009

### martyf

Re: 4-velocity

why?

9. May 11, 2009

### dx

Re: 4-velocity

What do you mean why? That's the definition of "timelike". A 4-vector is timelike if t² - x² - y² - z² > 0. All 4-velocites of ordinary particles must satisfy t² - x² - y² - z² = 1 (by definition), which is positive, so 4-velocities are timelike.

BTW, questions like this should be posted in the "Homework & Coursework Questions" forum.

Last edited: May 11, 2009
10. May 11, 2009

### Bob_for_short

Re: 4-velocity

Just calculate its square in the reference frame where the ordinary velocity is equal to zero.
Then only time-component contributes to the square. A time-like four-vector remains such in the other reference frames.

Bob.

11. May 11, 2009

### martyf

Re: 4-velocity

Definition of "timelike" : travel slower than light
WHY a 4-velocity that travel slower than light satisfies t² - x² - y² - z² = 1 ?!!

BTW, I'm trying to understand the meaning of time-like 4-velocity, I have not written any equation, any text of an exercise

12. May 11, 2009

### dx

Re: 4-velocity

Do you know the definition of 4-velocity?

13. May 11, 2009

### martyf

Re: 4-velocity

I wanted to say :

Definition of "timelike" 4-velocity :4-velocity of a PARTICLE THAT travels slower than light (ordinary particles). I wrote in a rush.
(if you're referring to this)

14. May 11, 2009

### dx

Re: 4-velocity

I'm just asking you what the definition of 4-velocity is. What are the components of the 4-velocity of a particle in a reference frame in which it's velocity is (vx, vy, vz)?

Last edited: May 11, 2009
15. May 11, 2009

### martyf

Re: 4-velocity

($$\frac{v_{x}}{c\sqrt{1-\beta^{2}}}$$ , $$\frac{v_{y}}{c\sqrt{1-\beta^{2}}}$$ , $$\frac{v_{z}}{c\sqrt{1-\beta^{2}}}$$ , $$\frac{i}{\sqrt{1-\beta^{2}}}$$ )

16. May 11, 2009

### dx

Re: 4-velocity

What's $i$ in the fourth component?

17. May 11, 2009

### martyf

Re: 4-velocity

complex number : i^2=-1

18. May 11, 2009

### ZikZak

Re: 4-velocity

Wow. Might you be able to find a textbook that was written after 1936? Because that was about the last time anyone ever used the ict convention. In that convention, the norm of the 4-velocity will always be -1, and is still timelike, because that is what timelike means in that convention. Get a newer book. Seriously. We've learned a lot since World War II.

19. May 11, 2009

### dx

Re: 4-velocity

No, 4-vectors don't have imaginary components.

The components of the 4-velocity uµ are

$$u_0 = \frac{1}{\sqrt{1-\beta^{2}}}$$

$$u_1 = \frac{v_{x}}{c\sqrt{1-\beta^{2}}}$$

$$u_2 = \frac{v_{y}}{c\sqrt{1-\beta^{2}}}$$

$$u_3 = \frac{v_{z}}{c\sqrt{1-\beta^{2}}}$$

Now, calculate $$u_0^2 - u_1^2 - u_2^2 - u_3^2$$. You should find that it is equal to 1.

20. May 11, 2009

### martyf

Re: 4-velocity

:) very funny, however my prof is able to find a textbook that was written after 1936. Infact I don't find a textbook that uses the ict convention!But I had to study on the prof's notes. The professors should be renovate their lessons, I should learn from them!