Groups | Search | Server Info | Keyboard shortcuts | Login | Register [http] [https] [nntp] [nntps]

Groups > sci.physics.relativity > #430856

Mass and kinetic energy (was: Talking Nonsense in Einstein's Schizophrenic World)

Show all headers | View raw

gehan.ameresekere@gmail.com wrote:
^^^^^^^^^^^^^^^^^^^^^^^^^^^
“Internet is the thing with cables; Usenet is the thing with *people*.”  
Your *real* *name* belongs there.  Subscribe to the newsgroup
or (better) use a proper newsreader to make that possible.

> On Monday, August 28, 2017 at 5:13:14 PM UTC+5:30, Gary Harnagel wrote:
>> On Monday, August 28, 2017 at 4:27:23 AM UTC-6, gehan.am...@gmail.com
>> wrote:
>> > On Sunday, August 27, 2017 at 9:45:59 PM UTC+5:30, Pentcho Valev quote:
>> > > Lisa Randall, Michio Kaku, Brian Cox, Neil deGrasse Tyson, Brian
>> > > Greene: "Now, listen carefully. The faster you move, the heavier you
>> > > get.
>> > "Now, listen carefully. The faster you move, the heavier you get"
>> > Relative to what?
>> 
>> This is shortcut talk.

ACK.

>> The faster an object moves (relative to an observer),
>> the more kinetic energy (relative to the observer), right? 

Correct.

>> By E = mc^2, or m = E/c^2, the more mass the object appears to have.

(In the following I am using the term “massive object” to mean an object 
whose mass is greater than zero.)

No, this is a common misconception.  E = m c² is the *wrong* equation if E 
is to be the kinetic energy or the total energy of a massive object and m is 
the object’s mass.

It is only correct if E *either* is the *rest* energy, *or* m is the so-
called “relativistic mass”: either E₀ = m c²; or E = m_{rel} c² with m_{rel} 
= γ m₀, where m₀ is the “rest mass”.  The latter two are ill-conceived, 
outdated concepts introduced by people other than Einstein – even explicitly 
condemned by Einstein –, and (as we can see here) potentially confusing to 
students; however, some scientists, like Michio Kaku from whom the sound 
bite is, are still willing to employ them – in popular-scientific talk at 
least.

In modern physics, there is only (rest) mass (m = m₀).  It is Lorentz-
*invariant* (it is *the same* for all observers, no matter their motion 
relative to the object) – as the Minkowski norm squared of the four-momentum 
is (*always*)

  ||P||² = −m² c²

(see below), and c is the isotropic speed of light –, and so is rest energy, 
but *kinetic* energy is not.
 
>> > How can you move faster compared to empty space which is nothingness?
>> Empty space is not nothingness.  For one thing, it contains vacuum
>> energy.

True, but irrelevant.  The correct, simpler explanation (that does not 
require either the theories of relativity nor quantum mechanics) is that 
velocity is *defined* as the change of position per unit time – v⃗ = ∂x⃗∕∂t 
–, and a frame of reference is required in order to define positions and 
times.  So speed, the magnitude of velocity, is *also* always *only* defined 
relative to something else (e.g., if you are sitting in a train that is 
moving relative to the ground, the speed of the train relative *to you* is 
*zero*).
 
> So the maximum Kinetic Energy an object can have relative to any observer
> is 1/2 mv^2 and where v=c, then 1/2 mc^2  ?

No (see also my other follow-up).  The kinetic energy is

  Eₖ = m c² (γ − 1)

with the Lorentz factor

  γ = 1∕√(1 − v²∕c²)

instead (in this notation you can most easily see that at relative rest, 
where v = 0 → γ = 1, the kinetic energy of a massive object is zero, as
it should be).

It turns out that

  Eₖ_{Newton} = 1∕2 m v²

is only a first-order approximation that only works for motion of an object 
with a relative speed that is much less than the speed of light (v ≪ c; 
which is why AFAIK nobody doubted it until Einstein posited special 
relativity).

In special relativity, the total energy E of a massive object can be derived 
from the norm of its 4-momentum:

   P⃗    = (E∕c, p⃗)

 ||P⃗||² = −(E∕c)² + p² = −(m c)²
          E²           = m² c⁴ + p² c²           Energy–momentum relation
          E            = √((m c²)² + (p c)²)
          E            = √((m c²)² + (γ m v c)²).

where P⃗ is the 4-momentum, and p⃗ is the 3-momentum.

Expanding γ as defined above, this can be written as

   E = γ m c².
     = m c²∕√(1 − v²∕c²),

(left as an exercise to the reader) and one can clearly see that the total 
energy at relative rest (rest energy; at v = 0) is

  E₀ = m c²

(as derived by Einstein before he found the energy–momentum relation, from 
Maxwell’s equations, in “Does the Inertia of a Body Depend upon its Energy 
Content?” [1905]).  [One can also see it immediately by setting p = γ m v
= 0 as v = 0 in the energy–momentum relation above.]

Why is this relevant?  Because the total energy of *any* object at *any* 
relative speed must be the sum of its *rest energy* and its kinetic energy:

  E   = E₀ + Eₖ.

So the kinetic energy for a massive object is

  Eₖ  = E      − E₀
      = γ m c² − m c².

Writing this as

  Eₖ  = 1∕√(1 − v²∕c²) m c² − m c²
      = (1 − v²∕c²)^(−1∕2) m c² − m c²

one can use binomial approximation (I learned it as “binomial expansion”) to 
see that the classical kinetic energy is *a good approximation* *only* as 
long as v ≪ c, i.e. v²∕c² ≪ 1:

  (1 ± x)^a ≈ 1 ± a x  where x ≪ 1,

so

  Eₖ = (1 − v²∕c²)^(−1∕2) m c² − m c²
     ≈ (1 + 1∕2 v²∕c²) m c²    − m c²
     = m c² + 1∕2 m v²         − m c²

  Eₖ ≈ 1∕2 m v² = Eₖ_{Newton}. ∎

<https://en.wikipedia.org/wiki/Kinetic_energy#Relativistic_kinetic_energy_of_rigid_bodies>

-- 
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.

Back to sci.physics.relativity | Previous | Next — Previous in thread | Next in thread | Find similar | Unroll thread

Thread

Talking Nonsense in Einstein's Schizophrenic World Pentcho Valev <pvalev@yahoo.com> - 2017-08-27 01:18 -0700
  Re: Talking Nonsense in Einstein's Schizophrenic World Bídníkùm Laterá <bila@wehia.ui> - 2017-08-27 09:52 +0000
  Re: Talking Nonsense in Einstein's Schizophrenic World Pentcho Valev <pvalev@yahoo.com> - 2017-08-27 09:15 -0700
    Re: Talking Nonsense in Einstein's Schizophrenic World Pentcho Valev <pvalev@yahoo.com> - 2017-08-28 00:55 -0700
      Re: Talking Nonsense in Pentcho's Schizophrenic World Gary Harnagel <hitlong@yahoo.com> - 2017-08-28 03:22 -0700
      Re: Talking Nonsense in Einstein's Schizophrenic World gehan.ameresekere@gmail.com - 2017-08-28 03:23 -0700
    Re: Talking Nonsense in Einstein's Schizophrenic World gehan.ameresekere@gmail.com - 2017-08-28 03:27 -0700
      Re: Talking Nonsense in Einstein's Schizophrenic World Gary Harnagel <hitlong@yahoo.com> - 2017-08-28 04:43 -0700
        Re: Talking Nonsense in Einstein's Schizophrenic World gehan.ameresekere@gmail.com - 2017-09-01 03:44 -0700
          Re: Talking Nonsense in Einstein's Schizophrenic World Gary Harnagel <hitlong@yahoo.com> - 2017-09-01 12:07 -0700
          Re: Talking Nonsense in Einstein's Schizophrenic World Odd Bodkin <bodkinodd@gmail.com> - 2017-09-01 14:53 -0500
          Mass and kinetic energy (was: Talking Nonsense in Einstein's Schizophrenic World) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2017-09-03 00:04 +0200
            Re: Mass and kinetic energy (was: Talking Nonsense in Einstein's Schizophrenic World) Gary Harnagel <hitlong@yahoo.com> - 2017-09-02 16:28 -0700
              Re: Mass and kinetic energy Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2017-09-03 02:11 +0200
                Re: Mass and kinetic energy Gary Harnagel <hitlong@yahoo.com> - 2017-09-02 17:29 -0700
                Re: Mass and kinetic energy Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2017-09-03 21:42 +0200
                Re: Mass and kinetic energy Gary Harnagel <hitlong@yahoo.com> - 2017-09-03 15:48 -0700
                Re: Mass and kinetic energy Ethel Ocallaghan <unlgclo@awlwaac.ag> - 2017-09-04 16:38 +0000
                Re: Mass and kinetic energy Ethel Ocallaghan <unlgclo@awlwaac.ag> - 2017-09-04 16:44 +0000
                Re: Mass and kinetic energy Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2017-09-05 08:37 +0200
                Re: Mass and kinetic energy Istrátora Metiku <untw@nwpovu.un> - 2017-09-06 12:58 +0000
            Mass–speed relation (was: Mass and kinetic energy) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2017-09-09 21:51 +0200
              Re: Mass–speed relation (was: Mass and kinetic energy) Utorských Vitelé <hue@svwor.hh> - 2017-09-09 21:12 +0000
        Re: Talking Nonsense in Einstein's Schizophrenic World gehan.ameresekere@gmail.com - 2017-09-01 03:46 -0700
          Re: Talking Nonsense in Einstein's Schizophrenic World Ned Latham <nedlatham@woden.valhalla.oz> - 2017-09-01 06:02 -0500
          Vacuum energy (was: Talking Nonsense in Einstein's Schizophrenic World) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2017-09-03 02:23 +0200
  Re: Talking Nonsense in Einstein's Schizophrenic World Pentcho Valev <pvalev@yahoo.com> - 2017-08-28 09:28 -0700
    Re: Talking Nonsense in Einstein's Schizophrenic World JanPB <filmart@gmail.com> - 2017-08-30 16:40 -0700
  Re: Talking Nonsense in Einstein's Schizophrenic World JanPB <filmart@gmail.com> - 2017-09-02 23:26 -0700

csiph-web