# B A couple of questions on mass increase

1. Mar 11, 2016

### alba

I found these [experimental data](https://en.wikipedia.org/wiki/Tests_of_relativistic_energy_and_momentum#Bertozzi_experiment) on mass increase.

> Data of the Bertozzi experiment show close agreement with special
> relativity. Kinetic energy of five electron runs: 0.5, 1, 1.5, 4.5, 15
> MeV (or 1, 2, 3, 9, 30 in mc²). Speed: 0.752, 0.828, 0.922, 0.974, 1.0
> in c (or 0.867, 0.910, 0.960, 0.987, 1 in c²).

- Do you know of any link where I can find more precise experimental data (say accurate to five digits) for mass increase?
For example, in the above table, is 0.5 MeV a rounded figure for 0.511 MeV , the rest mass of an electron? supposing the latter, what is the exact mass increase, is it exacly one mass (+ the usual rest mass), so the total mass is 2.00000 $m_e$?
And, if we give energy equal to 30 m(e) speed is surely not 1, what is the exact value?
- At what speed/energy there is minimum agreement with the SR formula? according to the picture it is about 5-6 MeV, is this correct?

- Can you also specify if "increased mass" means that the body exerts an increased (and exactly proportional) gravitational force? Does it mean that an electron with 1 GeV Ke exerts a pull equal to a proton?

2. Mar 12, 2016

### Orodruin

Staff Emeritus
You really should stop using "relativistic mass" as most physicists have abandoned the concept as redundant. You will still find it in popular science and older textbooks, but it really adds nothing to the discussion and tends to confuse more than it helps - one of the more common misconceptions being that the gravity of the object changes according to the relativistic mass (it does change, just not in accordance to relativistic mass).

3. Mar 12, 2016

### alba

I only mentioned mass increase, is that wrong? did I miss something in your link, does it answer my questions?

4. Mar 12, 2016

### Orodruin

Staff Emeritus
"Mass increase" only occurs if you use the concept of relativistic mass. An object's invariant mass (generally just called "mass" because we no longer use relativistic mass) is an invariant and the same regardless of the state of motion.

5. Mar 12, 2016

### alba

Could you re-frame my question in current appropriate terms, and reply to it? My main question is:

what happens if an electron gets 0.511 MeV of energy (or twice or 30 times as much), what is its speed and what is its mass (if it increases, doesn't it?) ,according to recent experiments, and how much they differ from the SR formula.

Thanks

6. Mar 12, 2016

### Orodruin

Staff Emeritus
You can easily compute the speed according to the relativistic relation between speed and total energy. All results are compatible with SR within experimental errors.

7. Mar 12, 2016

### alba

I know results are compatible with theory, what I am asking is to know the exact difference between theory and experiment in the range between 1 an 20-30 M(e) and where the disagreement is greatest. Do you know or can you direct me to a link?

8. Mar 12, 2016

### Orodruin

Staff Emeritus
There is no disagreement. Did you not read what I just wrote?

9. Mar 12, 2016

### alba

Is it possible to have some up-to-date figures or a chart like the one in the wiki article I quoted? can you give me at least one single figure, say,at 5 mc^2, please?

10. Mar 12, 2016

### Orodruin

Staff Emeritus
I do not understand what you are after. The agreement with special relativity is exceptional and modern particle accelerators would not function if this was not the case (not older ones either btw - old fashioned TVs needed to take relativistic effects into account).

11. Mar 12, 2016

### alba

You keep talking as if I am denying the agreement.

It is so hard to understand I am just asking for a list of figures more recent and accurate than Bertozzi's. (the one I quoted)? What is so strange about it?

Can I ask for the most precise figure avalaible of the speed of an electron with Ke of 9*0.511 Mev? is it exactly 0.974 c? THAT is what I am after!

Last edited: Mar 12, 2016
12. Mar 12, 2016

### Staff: Mentor

High-energy particle accelerators like the LHC are designed, and the experiments using them are analyzed, under the assumption that energy and momentum vary with speed in the way that relativity predicts. If the relativistic energy and momentum equations didn't work for proton energies up to 6.5 TeV at least, particle physicists would have noticed it by now.

13. Mar 12, 2016

### Orodruin

Staff Emeritus
You may rest asured that the agreement is good to a large number of significant digits. I do not know where you would find recent data, because this agreement is so good that things we have built would simply not work if it was not.

14. Mar 12, 2016

### alba

Can you tell me what is the greatest energy and highest speed ever reache in such facilitty? Is there a list of data of such experiments there?

15. Mar 12, 2016

### Staff: Mentor

The Large Hadron Collider at CERN has been accelerating protons to 6.5 TeV (6.5 x 106 MeV) since last year. That is the current record.

16. Mar 12, 2016

### alba

Why do you keep reassuring me, sir? haven't I made myself clear yet?
The chart I cited shows that agreement does vary at different energies, is that Bertozzi wrong? Has that experiment been disproved? Can you provide more recent or accurate data? If you can't , just say so. You haven't tried to answer one single question in my posts

17. Mar 12, 2016

### alba

Thanks, Do you have info regarding electrons, please? If you can't direct me to a link, please tell me the highest energy recorded and the exact speed corresponding to that, please

18. Mar 12, 2016

### Staff: Mentor

19. Mar 12, 2016

### Orodruin

Staff Emeritus
No, you misinterpret the result. In actuality, the Bertozzi experiment shows good agreement with SR within experimental uncertainty (you cannot expect more than that) and rules out the Newtonian description. That should be your takeaway message.

20. Mar 12, 2016

### DrGreg

21. Mar 12, 2016

### alba

Thanks for your link, but I found no data there, just a list of references Icannot access (including bertozzi):

Electron Relativistic Mass Variations
In the early 20th century there was an alternative theory by Abraham that is now little known, because these experiments rejected it in favor of SR. A critical review of the experimental evidence concerning the Lorentz model compared to the Abraham model was given in: Farago and Jannossy, Il Nuovo Cim. Vol5, No 6, pg 1411 (1957).
W. Kaufmann, Nachr. K. Ges. Wiss. Goettingen 2, pg 143 (1901) W. Kaufmann, Nachr. K. Ges. Wiss. Goettingen 3, pg 291 (1902); W. Kaufmann “Die elektromagnetische Masse des Elektrons”, Phys. Zeitschr. 4, pg 54 (1902) W. Kaufmann, Nachr. K. Ges. Wiss. Goettingen 4, pg 90 (1903) W. Kaufmann, “Uber die Konstitution des Elektrons”, Ann. Physik 19 ,495 (1906) and Nachtrag 20, 639–640 (1906); W. Kaufmann, “Uber die Konstitution des Elektrons”, Sitzungsberichte der preussichen Akademie der Wissenschaften, 1905, Part 2. W. Kaufmann, “Uber die Konstitution des Elektrons” Ann. Physik 19 ,495 (1906); W. Kaufmann, “Uber die Konstitution des Elektrons”, Sitzungsberichte der preussichen Akademie der Wissenschaften, 1915, Part A.H. Bucherer, Phyz. Zeitschr. 9 (1908), pg 755; Ber. d. deutschen Phys. Ges. 6 (1908), pg 688. A. Bucherer, “Die experimentelle Bestatigung des Relativitatsprinzips”, Annalen der Physik, 28, 1909E. Hupka, Ann. Phys. 31 (1910), pg 169 Cl. Schaefer and G. Neumann, Phys. Zeitschr. 14 (1913), pg 1117. G. Neumann, “Die träge Masse schnell bewegter Elektronen”, Ann. Phys. 45, pg 529 (1914) Ch.E. Guye and Ch. Lavanchy, Comptes rendus 161 (1915), pg 52Zahn and Spees, Phys. Rev. 53 (1938), pg 511Rogers et al., Physical Review 57 (1940), pg 37 Measurement of m/e and v for three beta-particles (electrons) from Radium. Supports the Lorentz model over the Abraham model by > 10 σ
• W. Bertozzi, Am. J. Phys. 32, 551 (1964).
Measurements of speed vs. energy for 0.5–15 MeV electrons.

22. Mar 12, 2016

### Staff: Mentor

Are you looking for something like a plot of v as a function of E with error bars?

23. Mar 12, 2016

### Mister T

No. It is much closer to 0.995 c.

You cannot have an exact value. There is always an uncertainty associated with every measurement. The rest energy of an electron is not exactly 0.511 MeV.

But, ignoring that for now, let's answer a different question that is as close to the one you're asking as I can come up with, and that has an exact answer. What is the speed of an electron whose kinetic energy is 9 times its rest energy? That electron would have a total energy that is 10 times its rest energy, because total energy equals kinetic energy plus rest energy, so $\gamma$ equals 10.

(Using the antiquated notion of relativistic mass, you would say that such an electron is 10 times heavier than when at rest, but instead it is a more modern usage to say that the total energy is 10 times the rest energy, where rest energy is equivalent to mass. And by mass I mean the ordinary mass, what you might call the rest mass to distinguish it from the relativistic mass.)

Anyway, since $\gamma=(1-\frac{v^2}{c^2})^{-1/2}$ the speed $v$ would be exactly $\sqrt{0.99}c$, or about 0.995$c$.

Modern particle accelerators achieve such high particle energies that it doesn't even make sense to look at the speed, because it's so close to $c$. Or equivalently, the kinetic energy is such a large fraction of the total energy that the rest energy is negligible. The distinctions made in the references you cited have all been been put to bed by modern experiments that impart energies that are orders of magnitude greater.

24. Mar 12, 2016

### alba

Yes, that is exactly what I am looking for
,https://en.wikipedia.org/wiki/File:BertozziExp.svg
something like the old one from 1964 I quoted, (bartozzi) with more precise and up to date figures. Surely those data can't be accurate since they were taken on such a short distance. I remarked above that such plot clearly indicate differnt agreements at different energies

Recent experiments am modern colliders, moreover, test energies much greater than 15 MeV, right?

Last edited: Mar 12, 2016
25. Mar 12, 2016

### alba

I am not asking for an exact value, but for an appproximation of 5 digits and surely an experiment at LHC or the like produces such approximation.

- The predicted value for an increase of 9 rest masses is v = 0.994 987 437 (2) with the accuracy of 9 digits or, if we take all 10 digits, we get the value od 10.000 000 09 masses)
- At LHC or at any other sinchrotron, you know the exact energy provided, the magnetic field and the radius r of the collider

I suppose the can get a result with a five-digit accuracy, am I wrong?