Inertial Frames: Constant Quantities?

[Zahid Iftikhar]

My question is about some physical quantities which two observers in two respective inertial frames will find the same. I wonder are there any such quantities? Some books say force, speed of light etc are constants for both the observers. Please guide me on this.
Regards.

 

[PeroK]

My question is about some physical quantities which two observers in two respective inertial frames will find the same. I wonder are there any such quantities? Some books say force, speed of light etc are constants for both the observers. Please guide me on this.
Regards.

The term you are looking for is invariant quantities.

The speed of light, $c$ is one - often the defining one.

The spacetime distance between two events is invariant: $-c^2(\Delta t)^2 + (\Delta x)^2$.

In general, the length of any four-vector is an invariant. Spacetime distance is the length of the displacement four-vector.

Another good example is the energy-momentum four-vector, in which case the invariant quantity is the invariant mass:

$m^2c^4 = E^2 - p^2c^2$

All good stuff!

 

[Dale]

In the literature these are called invariants. These include mass, proper time, proper acceleration, phase, $E^2-B^2$, $\rho^2-J^2$, and of course c.

Edit: I am both too little and too late!

 

[weirdoguy]

Force is the same in Newtonian mechanics, but it changes in relativity.

 

[robphy]

The relative speed of the other frame.

 

[pervect]

Something I wanted to clarify. Force, by which I mean the traditional 3-component vector or 3-force, is definitely not an invariant in special relativity. An invariant force-related entity can be created by adding an an additional component to the 3-force vector to make it a 4-force vector. This 4-vector is in and of itself not invariant in the technical sense, because the components that make it up change when one changes ones frame of reference. However, the magnitude, or length, of the 4-vector is a true invariant, which is the same for all observers. The 4-force vector also transforms in an identical way to other 4-vectors, so while one may have to do some work and reading to understand 4-vectors, once one does so they have many different applications.

 

[SiennaTheGr8]

Something I wanted to clarify. Force, by which I mean the traditional 3-component vector or 3-force, is definitely not an invariant in special relativity.

I hesitate to mention this, given the level of the thread, but it also turns out that 3-force (magnitude) is Lorentz-invariant in the one-dimensional case. (More generally, the Cartesian component of 3-force that's parallel to the axis of a Lorentz boost is invariant under said boost. The other components aren't.) I don't attach much meaning to this, though.

 

[Nugatory]

I hesitate to mention this, given the level of the thread, but it also turns out that 3-force (magnitude) is Lorentz-invariant in the one-dimensional case. (More generally, the Cartesian component of 3-force that's parallel to the axis of a Lorentz boost is invariant under said boost. The other components aren't.) I don't attach much meaning to this, though.

Seeing as how most intro treatments only consider cases in which the boost is aligned with the x-axis... I think this is worth mentioning.

 

[Zahid Iftikhar]

Thanks indeed to all worthy scholars for their time. I have very basic knowledge of relativity. Understanding 4D analyses seems hard. What if I say in 2D motion if relative motion of the frames is along x-axis, can we say the observers in both frames will agree upon the width (length along y-axis perpendicular to direction of motion) of all the objects inside both the frames to be invariant?

 

[Zahid Iftikhar]

The relative speed of the other frame.

Thanks for the reply. Do you mean both the observers will measure same speed of each other's frame of reference? But speed is length divided by time. If length shrinks and time dilates, should the speed not change? My explanation may reveal my very basic knowledge of relativity. I apologize for it.

 

[Zahid Iftikhar]

In the literature these are called invariants. These include mass, proper time, proper acceleration, phase, $E^2-B^2$, $\rho^2-J^2$, and of course c.

Edit: I am both too little and too late!

But mass increases and time dilates a per special theory of relativity. How are they invariant? please guide.
How is acceleration invariant? Please explain. Regards

 

[Zahid Iftikhar]

The term you are looking for is invariant quantities.

The speed of light, $c$ is one - often the defining one.

The spacetime distance between two events is invariant: $-c^2(\Delta t)^2 + (\Delta x)^2$.

In general, the length of any four-vector is an invariant. Spacetime distance is the length of the displacement four-vector.

Another good example is the energy-momentum four-vector, in which case the invariant quantity is the invariant mass:

$m^2c^4 = E^2 - p^2c^2$

All good stuff!

Thanks for the reply.
Are there any quantities in 2D or 3D invariant? please mention. Regards

 

[Dale]

But mass increases

The usual meaning of mass now is the quantity $m^2 c^2=E^2/c^2-p^2$. This quantity is invariant. There is some confusion due to an outdated concept called “relativistic mass” that has not been in use for decades among professional physicists.

and time dilates

Proper time does not dilate, that is why I specified proper time.

 

[Zahid Iftikhar]

The usual meaning of mass now is the quantity $m^2 c^2=E^2/c^2-p^2$. This quantity is invariant. There is some confusion due to an outdated concept called “relativistic mass” that has not been in use for decades among professional physicists.

Proper time does not dilate, that is why I specified proper time.

Thanks Sir. You mean the time in one's own frame of reference. I am often confused to understand this term. I apologize.

 

[Ibix]

What if I say in 2D motion if relative motion of the frames is along x-axis, can we say the observers in both frames will agree upon the width (length along y-axis perpendicular to direction of motion) of all the objects inside both the frames to be invariant?

Imagine a disc that just fits through a ring (like one of those babies' shape sorting toys). If width expands or contracts in different frames, describe what happens when the disc tries to pass through the ring, in the rest frame of the ring and in the rest frame of the disc.

Do you mean both the observers will measure same speed of each other's frame of reference?

Yes. Imagine two identical cars in a head-on collision. In car 1's rest frame, all the energy that damaged the cars came from car 2's kinetic energy. In car 2's frame it all came from car 1's kinetic energy. But they agree on the damage done, and how much energy that must have taken. So what does that tell you about the kinetic energy (and hence velocity) of car 2 in car 1's frame and of car 1 in car 2's frame?

But speed is length divided by time. If length shrinks and time dilates, should the speed not change?

There's a third effect called the relativity of simultaneity, which is critical to understanding relativity. Your analysis is incorrect because you haven't included it. Look up the Lorentz transforms, and then apply them to one car measuring the other's speed and vice versa.

But mass increases and time dilates a per special theory of relativity. How are they invariant? please guide.

Relativistic mass increases with velocity, but serious sources largely stopped using the term decades ago because it causes nothing but confusion. Dale is referring to rest mass (also known as invariant mass), and this is what is usually meant by "mass" these days. It doesn't change.

Your proper time is the time your wristwatch measures. My proper time is the time my wristwatch measures. If we meet up and synchronise watches, then go off and do different things and meet up again, our proper times won't necessarily agree. But our frames must describe the other's proper time as an invariant - otherwise I'd predict your watch reading incorrectly. Relativity would be wrong if it couldn't predict your watch reading.

More generally, all instrument readings must be invariants. Although frames won't necessarily agree why the instruments read what they do, they must agree the readings.

How is acceleration invariant?

Because proper acceleration is something you can measure in a closed box - for example you feel yourself pressed back into the seat when you accelerate in a car. That's a (crude) instrument for detecting proper acceleration. Thus everyone must agree what acceleration you feel - or else we can't explain why you are pressed into your seat.

Are there any quantities in 2D or 3D invariant?

The 2d case is just a 4d case where you've required two of all your vector components to be zero. So anything invariant in the general case is also invariant in the 2d or 3d case. As @SiennaTheGr8 points out, you get some extra invariants in 2d.

 

[Zahid Iftikhar]

Thank you very much all very helpful scholars. After reading all these responses I inwardly realize I need to go through basic lessons of theory of relativity. Would you suggest me some sources where a novice may learn it.
High regards.

 

[PeroK]

Thank you very much all very helpful scholars. After reading all these responses I inwardly realize I need to go through basic lessons of theory of relativity. Would you suggest me some sources where a novice may learn it.
High regards.

I would recommend:

https://www.goodreads.com/book/show/6453378-special-relativity

 

[SiennaTheGr8]

Thank you very much all very helpful scholars. After reading all these responses I inwardly realize I need to go through basic lessons of theory of relativity. Would you suggest me some sources where a novice may learn it.
High regards.

Two recommendations:

1) Spacetime Physics by Edwin Taylor and John Wheeler (a classic)

2) Special Relativity for the Enthusiastic Beginner by David Morin (a new one that does a great job introducing the relativity of simultaneity)