Understanding Ultrarelativistic Speeds and Their Boundaries

[MathematicalPhysicist]

So I read the Wiki entry, and from what i gather ultrarelativitic speeds are speeds that almost the speed of light (I mean $$\gamma >>1$$ is the same as $$v \to c$$).

Now what is the boundary where where you regard a particle's speed as ultrarelativitic compare to nonrelativitc speeds.

Is $$v \geq 0.9c$$ regarded as ultra (I guess that it is), but then what about speeds such as 0.8c, 0.7c etc?

My naive notion was that the lowest boundary is 0.5c that below this we would have nonrelativitic speeds, but I am not sure.

 

[jtbell]

There's no "official" definition for "nonrelativistic speed." To me, it means, "slow enough that relativistic equations give results indistinguishable from classical equations, for whatever amount of accuracy you want." You have to decide whether you want 1% accuracy, or 0.1% accuracy, etc.

 

[MathematicalPhysicist]

Ah, Ok.

How do you decide what type of accuracy you need, can you give an experimental example?

It seems quite arbitrary, doesn't it?

I wonder also for technological reasons, cause I assume that engineers use relativistic corrections.

 

[Nugatory]

Ah, Ok.
How do you decide what type of accuracy you need, can you give an experimental example?

We make relativistic corrections for particles in a particle accelerator, but not for billiard balls on a billiard table.

It seems quite arbitrary, doesn't it?

No more arbitrary than choosing to measure the distance between cities to the nearest kilometer but the distance between features on an integrated circuit to the nearest nanometer.

 

[Nugatory]

cause I assume that engineers use relativistic corrections.

Very very seldom. The engineering of the GPS system and large particle accelerators requires relativistic corrections, but there aren't many more examples.

 

[PeterDonis]

So I read the Wiki entry, and from what i gather ultrarelativitic speeds are speeds that almost the speed of light (I mean $$\gamma >>1$$ is the same as $$v \to c$$).

Some people divide things into three categories, based on momentum rather than speed; something like this, where $p$ is momentum and $m$ is rest mass (using units where c = 1, so momentum and mass have the same units):

Nonrelativistic: $p << m$

Relativistic: $p \approx m$

Ultrarelativistic: $p >> m$

Since $p = \gamma m v$, we can express this in terms of speed $v$ as follows (using $\gamma = 1 / \sqrt{1 - v^2}$):

Nonrelativistic: $v << \sqrt{1 - v^2}$

Relativistic: $v \approx \sqrt{1 - v^2}$

Ultrarelativistic: $v >> \sqrt{1 - v^2}$

From this we can see that the "relativistic" regime is around $v = 1 / \sqrt{2} \approx 0.707$; speeds much smaller than that are nonrelativistic, and speeds much larger than that are ultrarelativistic.

 

[WannabeNewton]

How do you decide what type of accuracy you need, can you give an experimental example?

You look at the velocity scale of the system using dimensionless quantities. For GR and SR purposes you can use fundamental constants, characteristic time scales and characteristic length scales to build dimensionless constants $\hat{c}$ and $\hat{G}$ that respectively characterize the velocity scale and scale of self-gravitation of a system.

It seems quite arbitrary, doesn't it?

No it comes right out of the velocity scale of the system. Characteristic length scales and time scales and characteristic couplings and all scales derived from them dictate all of our approximation regimes. This is ubiquitous throughout physics although conceptually complicated in QM and QFT whereas conceptually simple in classical physics.

 

[ghwellsjr]

So I read the Wiki entry, and from what i gather ultrarelativitic speeds...

I did a search on "ultrarelativitic speed" in wikipedia and it said "There were no results matching the query".

 

[jtbell]

For "ultrarelativistic" I tend to think in terms of energy, because my background is in particle physics. If the rest-energy E₀ of a particle (corresponding to its rest mass via E₀ = m₀c²) is much smaller than its kinetic energy (or also its total energy, kinetic + rest), then I consider it to be "ultrarelativistic" rather than merely "relativistic."

A proton has a rest-energy of about 1 GeV, so I would consider a proton with 5 or 10 GeV energy to be "relativistic", but one with 100 or 1000 GeV to be "ultrarelativistic."