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What is the maximum velocity that can be measured between two objects? Is $2c$ the correct answer?

Two photons (A & B) are emitted simultaneously from my position; photon A going north and photon B south. Both photons are traveling in opposite directions such that a single straight line can be drawn through A, B, and our position.

Given:

  1. Photon A is moving north with a velocity $v = c$

  2. Photon B is moving south with a velocity $v = c$

  3. The rate of increase of the distance between A and B, expressed as a velocity, is $v = 2c$

Therefore, the maximum possible measurable velocity between two objects = $2c$

BLAZE
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PatrickH
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  • Related: http://physics.stackexchange.com/q/11398/2451 and links therein. – Qmechanic Mar 01 '16 at 22:22
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    You can measure the relative velocity of two objects to be up to $2c$, yes. But the relative velocity of an object to yourself is at most $c$. – knzhou Mar 01 '16 at 22:26
  • @knzhou Don't use the term relative velocity when looking at the change of separation distance/time for two different objects moving relative to a reference frame. The relative velocity of the two objects must use the Lorentz transformation. I agree, the $\Delta d/\Delta t$ ratio can be larger than $c$, but that is NOT the relative velocity of one object to another. – Bill N Mar 01 '16 at 23:15
  • @BillN I agree my use of the term is misleading, but what would you call it, in words? – knzhou Mar 01 '16 at 23:19
  • It doesn't need a name. I describe it as the ratio of separation change to time, or the time rate of change of separation distance. – Bill N Mar 01 '16 at 23:21
  • "Velocity differential" seems like a good candidate. – Alex Meiburg Mar 01 '16 at 23:39
  • "but what would you call it, in words?" - closing speed. – Alfred Centauri Mar 02 '16 at 00:16
  • Several answers/comments confirm that the greatest MEASURABLE velocity/speed is 2c. I just don't beleive that statement can be true unless someone can convincingly explain how 2c can actually be MEASURED! – Jens Mar 03 '16 at 21:19

4 Answers4

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Consider two objects approaching each other, along a line through both objects, with speed $u_1$ and $u_2$ respectively in some inertial frame of reference (IRF).

The distance between the two objects, decreases at the rate of $u_1 + u_2$; this is the closing speed. Note that this speed is not a relative velocity and so the relativistic velocity addition formula does not apply.

Thus, for massive objects, this closing speed can approach arbitrarily close to $2c$.

However, from the perspective of either object, the relative velocity of the other object is less than $c$ since the relativistic velocity addition formula applies.

In summary, the rate of decrease of the distance between two objects, as measured in some IRF, is not the speed of any object and thus, is not limited to $c$ or less. But, since the relative speed of any object cannot exceed $c$ (as far as we know), the closing speed cannot exceed $2c$.

Alfred Centauri
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    I just receive my test results back and Alfred's answer and reasoning agree with the instructor and my answer. To let you know Alfred, your explanation is a lot clearer than my instructors. Thanks – PatrickH Mar 04 '16 at 17:21
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This is really a plain old Special Relativity question about addition of velocities. The guy in the middle gave object 1 a velocity $-v_1$ and object $2$ a velocity $v_2$. When you ask for the relative velocity between the two objects, you mean as observed from sitting on one of the objects (say object 1). In this case, you see the guy in the middle receding from you with $v_1$. In his frame he throws object $2$ with $v_2$. You see the special relativistic addition of $v_1$ and $v_2$ which is always less than $c$. $$ \lambda_1 = \tanh^{-1}\left(\frac{v_1}{c}\right) $$ $$ \lambda_2 = \tanh^{-1}\left(\frac{v_2}{c}\right) $$ $$ \frac{v}{c}=\tanh\left(\lambda_1+\lambda_2\right) $$ For fun, this is shown using Lorentz Boost parameters which are additive when the velocities are in the same direction. The maximum tanh can become is $1$. Therefore $c$ is the maximum possible relative velocity, not $2c$.

BLAZE
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Gary Godfrey
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The velocity a body can show is $5c$ which can be obtained by going at $0.5^{1/2}c$ i.e one divided root two $c$. Going at more or less velocity causes you to show less velocity in the real world or the actual reference frame so max relative velocity is $0.5+0.5=1c$.

BLAZE
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istiaq
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Two objects/photons moving at c in opposite direction, the distance will increase at 2c for a "stationary" observer in between them.

But They can not see or communicate with one another. Therefore, their relative speed is immeasurable.

Suppose, they start with c, they will instantaneously disappear for each other.

kpv
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  • "Suppose, they start with > .5c to begin with, they will instantaneously disappear for each other." -- This is plain wrong. – Norbert Schuch Mar 02 '16 at 15:15
  • @ Norbert Schuch: Thanks for the correction. I changed .5c to c. Hope that is ok. – kpv Mar 02 '16 at 20:33
  • They can't move at c. – Norbert Schuch Mar 02 '16 at 21:11
  • Suppose A is moving directly towards B with speed .8c (according to the observer in the middle) and vice versa. Also the observer sees both A and B are emitting light of the same frequency. The observer should see a doppler shift upwards corresponding to .8c. Would A and B then see an upwards doppler shift corresponding to 1.6 c? – Jens Mar 02 '16 at 21:13
  • Thus if the doppler shift corresponds to 1.6 c then they should each conclude that the other is approaching with a speed substantially faster than light so time dilation (frequency lowering) must somehow have an influence to make things right. – Jens Mar 02 '16 at 21:25