What Is The Reference Frame For The Velocity Of An Object In The Universe?

As I understand it, movement of two objects is relative to one another. Velocity is always in relation of one thing to another. A car travels at 100km/h (60mi/h) relative to the rotating Earth surface it's driving on. A rocket propels upwards relative to the rotating Earth surface.

If we choose a different reference point, things change. If we use the center of the Earth as a reference point, the car, parked at the side of the road, is technically rotating with the Earth. If you were to drive east at the same speed as earth rotates, we would now be going twice as fast. And if we drove west at the same speed as the earth rotates, we would be stationary relative to the center of the Earth.

Similarly, if we launch a rocket in the direction Earth is rotating, at the same speed as Earth is moving around the sun, then relative to the sun it is moving twice as fast. If we launched the rocket in the opposite direction, then relative to the sun it wouldn't be moving (other than starting to fall straight into the sun).

Thus, there really isn't one speed for any object but multiple speeds depending on what you use as the frame of reference. And right now, even staying still, we are moving in all sorts of ways from the universe, the sun, the Earth around the sun, and the Earth's rotation.

We know that no object can travel at the speed of light. As an object goes to approach the speed of light, the amount of energy increases exponentially until it reaches infinity. However, a velocity must be in relation to something. That frame of reference must be defined somewhere.

If an object is already moving at half of the speed of light, and from it we launch a smaller object as fast as we possibly can, the speed of that object relative to us could only ever get to half the speed of light, because the total speed could never reach the speed of light.

And if we launched that smaller object in the other direction, could it not therefore move at up to 1.5 times the speed of light relative to our frame of reference?

From this then, we could see that the object we launched in the same direction would take substantially more energy to accelerate than the object launched in the opposite direction. Furthermore, if we slow down or speed up our spaceship, the relationship would change.

Could it not then be possible to launch objects in all direction and be able to determine the exact speed we are moving at, and from there, calculate what the reference frame of the universe is?

• You are standing on the ground. Alice is traveling eastward at speed $.5$ (that is, half the speed of light) relative to you. She launches objects eastward and westward at speeds relative to her of (say) $.9$. Then relative to you, the eastbound object travels at speed $.9655$ and the westbound object at speed $.2759$ (I got these by applying the relativistic velocity addition formula, which you can Google). Nothing stops her from launching these objects at any speed she wants between $0$ and $1$ relative to herself, so your scheme can't be used to find her "absolute" speed. – WillO Jul 27 '17 at 23:19
• PS---although this is a very elementary question, I do not think that the OP could reasonably have been expected to know what phrases to Google for, so I think the downvote is uncalled for. – WillO Jul 27 '17 at 23:20
• The kinetic energy of an object is also reference-frame-dependent. You also contradict yourself. You say that we know that objects can't move the speed of light, c, then you propose that something moves at 1.5c without showing any attempt to research special relativity velocity equations. Your logic doesn't flow well. – Bill N Jul 28 '17 at 0:35

In an expanding universe you have the peculiar velocity which is measured relative to the cosmic microwave background (in the frame of a comoving observer the CMB has the same wavelength in all directions, while an object with peculiar velocity sees a dipole in the CMB).

If you measure an object's velocity relative to a distant observer you also have to add the recessional velocity that can sure enough exceed the speed of light, which is of course allowed by general relativity.

So the total velocity between 2 objects is their peculiar velocities added like in special relativity and the recessional velocity between both added classicaly.

Example: if two objects are seperated by the distance of 2 hubble radii, the recessional velocity between both is +2c. If they locally travel toward each other with c/2 relative to the CMB, their special relativistic velocity toward each other is -(v₁+v₂)/(1-v₁·v₂/c²)=-4/5c. But since the recessional velocity between both is 2c, the total velocity is 2c-4/5c=+6/5c, so they still would move away from each other with more than c.

The first answer has the numbers right, and the rationale. This is just to explain the phrase in that answer 'the recessional velocities are added classically'. They really are not added classically but calculated from General Relativity (GR) and the answers are the same, just because the way the Hubble parameter is defined, and because both cosmological observers use the same reference frame, defined uniquely by the expansion of the universe and it's homogeneity. See next.

[as an aside, note that you added velocities classically, but the peculiar velocities need to be added as in the answer, using special relativity]

The GR equations lead to Hubble law which relate velocity to distance as

v = Hr

And thus we get the Hubble distance when v = c as c/H. That Hubble distance changes with time, as H is not a constant, but not too fast. Usually people call the Hubble distance the Hubble distance now, where H is the value of the Hubble parameter now. Note that the units of the Hubble parameter is velocity/distance, usually stated as 68 km/sec/Mpsec, for its value now, per the standard cosmological model at this point.

Note in https://en.m.wikipedia.org/wiki/Hubble%27s_law tHat the so called Hubble radius $(c/H)_{now}$ is about 14.4 billion light years, slightly larger than the universe radius now of about 13.8 billion light years. They are not the same number because of the way the expansion has and is occurring, that it is not an exactly linear expansion. This is according to the standard cosmology model.

The value of H now is denoted as $H_0$, meaning now. It is also called the Hubble parameter now. The Hubble radius is also called the Hubble horizon. See it and a number of other useful horizon definitions at https://en.m.wikipedia.org/wiki/List_of_cosmological_horizons

Next, the 68 Kms/sec/Mpsec means that if you are about 14.4 billion light years away from me your speed wrt me is and if another galaxy is another 14.4 billion light years from you their speed is c wrt you. And their speed wrt to me is 68 Kms/sec/Mpsec times 28.8 billion light years (sorry, you have to change Psec to light years, and the math comes out) comes out to be twice as before, or 2c. Both you and me would be using the comoving reference frame, and would agree on times. We both would have to measure those distances at cosmological time (= comoving time) now. It is a well defined frame of reference, where the CMB is homogeneous and isotropic, and where we would all agree on the age of the universe. Sso it winds up being additive for the velocities. But it is a non-trivial result, certainly not just adding them classically.