Meaning and derivation of 4-vector

In summary, the Lorentz transformation matrix defines the first column as -ct, not t itself, which satisfies the units of x, y, and z. This is because there are only a few geometries that have an invariant speed and obey a relativity principle, and the one where (ct)^2 picks up a minus sign is the best candidate for modeling our world. The 0th component of the four-momentum is the energy divided by c, and the scalar product for 4-vectors is defined as -(ct)^2 + x^2 + y^2 + z^2 due to the Lorentz transformation. This is different from the scalar product for 3-vectors, where it is x^2
  • #1
simoncks
29
0
Meaning of ct in Lorentz transformation -
In Lorentz transformation matrix, the first column is defined as - ct, not t itself. Is it because ct satisfies the units of x, y, z? Or, simpler Lorentz transformation matrix will be derived? The idea of 'ct', instead of t, is quite abstract for me. Not sure whether it is conceptually correct to consider ct as -
a meter of x*(0) corresponds to the time it takes light to travel 1 meter in vacuum.

Derivation and use of 4-vector scalar product -
Why is the scalar product defined in such way, where square of 'ct' has a minus sign? How to prove its invariant property for any inertial frame? Already in a big mystery...
How about the momentum 4-vector? What does p0 in the first row mean?

Thanks a lot.
 
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  • #2
A meter along the [itex]ct[/itex] axis corresponds to the time it takes for light to travel one meter, yes.

There are only a few geometries that have an invariant speed and obey a relativity principle. The geometry where [itex](ct)^2[/itex] picks up a minus sign happens to be the best candidate for modeling our world.

The 0th component of the four-momentum is the energy (divided by [itex]c[/itex]).
 
  • #3
simoncks said:
Meaning of ct in Lorentz transformation -
In Lorentz transformation matrix, the first vector is defined as - ct, not t itself.

Relativists typically work in units where c=1, so the distinction between t and ct is irrelevant.

I assume you mean the first component of the vector, rather than the first vector?

It's not true generically that the 0th component of the position vector has that minus sign. Are you working in a context where covariant and contravariant vectors like xk and xk are defined? Or are you talking about old-fashioned treatments where the timelike component is ict (not -ct)?
 
  • #4
There are only a few geometries that have an invariant speed and obey a relativity principle. The geometry where (ct)2 picks up a minus sign happens to be the best candidate for modeling our world.
Isn't the velocity of an object varying in different inertial frames? Can you explain further about why this geometry is the best candidate? It is quite abstract...

The 0th component of the four-momentum is the energy (divided by c).
Is there indeed a reason why 0th component is E/c? Or, it is simply defined in such way?

I assume you mean the first component of the vector, rather than the first vector?
Yes. Sorry, to be confusing.

It's not true generically that the 0th component of the position vector has that minus sign. Are you working in a context where covariant and contravariant vectors like xk and xk are defined? Or are you talking about old-fashioned treatments where the timelike component is ict (not -ct)?
I am working in covariant and contravariant vectors, but what confuse me is the scalar product. For 3-vector, the scalar product is x2+y2+z2. The 4-vector one is rather different -
-(ct)2+x2+y2+z2
It is a problem why the scalar product is defined such an 'abnormal' way, with a minus sign.
 
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  • #5
simoncks said:
I am working in covariant and contravariant vectors, but what confuse me is the scalar product. For 3-vector, the scalar product is x2+y2+z2. The 4-vector one is rather different -
-(ct)2+x2+y2+z2
It is a problem why the scalar product is defined such an 'abnormal' way, with a minus sign.

Does this help?

For a 3-vector, it's x2+y2+z2 that is the same for all observers, i.e., regardless of how you rotate the x, y, and z axes.

For a 4-vector, it's easiest to think in just one spacelike dimension and one timelike one, so we have only t and x. Then the Lorentz transformation looks like figure k here: http://www.lightandmatter.com/html_books/lm/ch23/ch23.html . What stays the same for all observers is -t2+x2 (in units with c=1). This is because the Lorentz transformation doesn't change the diagonal line x=t, which represents motion at the speed of light.
 
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  • #6
Thanks for your help. The link gives a really clear explanation.
 

FAQ: Meaning and derivation of 4-vector

What is a 4-vector?

A 4-vector is a mathematical object that has four components and obeys certain transformation rules under rotations and boosts in special relativity. It is commonly used in physics to describe the position, momentum, and energy of particles in spacetime.

What is the significance of the number 4 in a 4-vector?

The number 4 represents the four dimensions of spacetime - three dimensions for space and one dimension for time. A 4-vector incorporates both spatial and temporal information, making it a powerful tool for describing physical quantities in a relativistic context.

What is the difference between a 4-vector and a regular vector?

A regular vector has three components and represents a physical quantity in three-dimensional space. A 4-vector has four components and represents a physical quantity in four-dimensional spacetime. Additionally, the components of a 4-vector transform differently under rotations and boosts than those of a regular vector.

How is a 4-vector derived?

A 4-vector is derived from the principles of special relativity, which describe the relationships between space and time in the presence of relative motion. By incorporating these principles into the mathematics of vectors, a 4-vector is created that can accurately describe the behavior of particles in spacetime.

What are some common examples of 4-vectors in physics?

Some common examples of 4-vectors include the position 4-vector, which describes the position of an object in spacetime, and the energy-momentum 4-vector, which describes the energy and momentum of a particle. Other examples include the 4-velocity and 4-acceleration 4-vectors, which describe the motion of particles in spacetime.

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