Question of the Week

Is there an absolute maximum temperature?

Sun, 25th Mar 2012

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Bronwyn Higgs asked:

Dear Chris,


I have a question about absolute zero.


If we can have a lower limit on temperature -absolute zero -i.e. so cold that nothing happens, why do we not have an upper limit - i.e. a temperature that is so hot that atoms and particles are so hot that they are ripped into the most elementary units and become so excited that theycannot react with anything around them.


It has always seemed to me that there is a certain asymmetry about this temperature scale.


Thanks for taking questions from interested members of the public.



Bronwyn Higgs


We posed this question to Sam Gregson, High Energy Particle Physicist at the University of Cambridge... 

Sam -   The temperature of a system is simply related to the amount of energy in that system.  Because the system can't have a negative energy, there is only so much heat you can remove from it and so a limit to how cold you can get.  This is called absolutely zero.  We’ve got very close to it.  Scientists in Finland have cooled rhodium atoms to a 10th of a billionth of a degree above absolute zero.  On the other hand, an absolute maximum temperature would require there to be a limit to the amount of energy you canA thermometer give to a particle.  As far as we know, there is no such limit. 

Although the speed of light is the universal speed limit, the reason you can't get there is that this would require an infinite amount of energy.  So this speed limit does not limit the amount of energy and therefore, the temperature of an individual particle. 

The most energetic particle ever observed was a cosmic ray over Utah, travelling at 99.999999999985% of the speed of light.  Probably a single proton with about 50 joules of energy.  This is equivalent to about 5 trillion trillion degrees Celsius and there is no evidence that this is the hottest you could get to. 

As far as we know, you are just limited by the amounts of energy you can give to a particle.  So you could say that the absolute maximum temperature is a temperature equivalent to all the energy in the universe, concentrated onto one particle.  But that limits more accounting than basic physics.

Hannah -   Thanks, Sam Gregson from Cambridge and CERN.  So temperature is related to thermal energy and Einstein’s theory of relativity means that although a particle has a universal speed limit, it doesn’t have an energy limit.  If you took all of the energy in the universe and put it into one particle, you'd essentially run out of energy before you run out of capacity for energy which is why we have no absolute maximum temperature. 


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I thinik theres 2 aspects to this question:

1) The temperature at which fundamental sub-atomic particles get wrenched apart and cease to exist - thiere will be presumably a max temperature at which this happens
2) The availability of an energy source to create the heat. This is going to limit the max temperature available, and then begs the question - how is the energy released.
Like any good scientist, I have failed to answer the original question as well as posing two more of my own
Sprool, Tue, 20th Mar 2012

Temperature is a measure of how fast the elements of an object are moving - and the absolute speed limit is the speed of light - so it ought to follow that the absolute max temperature is when the particles are approaching the speed of light - like the protons at CERN in the LHC - they are within a few meters per second of the speed of light.

Apparently the LHC achieved temperatures of 1.6 trillion degrees Celsius - so this must be close to the top limit. chriswf88, Wed, 21st Mar 2012

But as things get faster they also get heavier (E=mc^2) - so as you get hotter then the energy needed to raise the temperature further goes up, so it should tend to infinity... chris, Wed, 21st Mar 2012

I think Chris is right.  To be more precise I think the second Chris is correct.  A maximum finite speed or velocity does not - in relativistic terms - cause a maximum finite energy.  As the Kinetic Theory of Gas derived temperature is related to the average Kinetic Energy (rather than the average speed) of particles - at non-relativistic speeds

E_{k} = \frac{1}{2}mv^2

E_{k} being the kinetic energy.

\bar{E}_{k} = \frac{1}{2}kT

Where \bar{E}_{k} is the average Kinetic Energy per degree of freedom.

However at relativistic speed the Kinetic Energy changes - the easiest way of thinking about it is the total energy minus a rest mass energy
E_{k} = mc^2 -m_{0}c^2

But we can get rid of the pesky different forms of mass (which are horrible)

E_{k} = \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}} -mc^2

as v increases the denominator of the fraction becomes smaller and smaller (tending towards but not reaching zero) and thus the first term gets bigger and bigger and the energy does the same.  But after writing all that out I realise I am not sure that it is legitimate just to plug a relativistic formula for KE into the standard formula for temperature.  If is is legitimate it shows that second Chris is correct....

all the stuff I can find however uses the alternative definition of temperature (via entropy, degrees of freedom etc ) which I think would be a better route

imatfaal, Thu, 22nd Mar 2012

I look forward to the answer on the next show.

I hope that whatever the solution it also might address the possible temperature of a singularity or even the big bang as there must have been a fair amount of kinetic energy stored up in that little spec of mass just before it all started to rip space time apart!  If there is a maximum then it must have been pre bang?

I just read an article about the 4 trillion degrees achieved by a team in 2010, they state that the predicted temperature for  photons and protons to melt is 2 trillion degrees.

The LHC got to 10 trillion degrees at the end of 2010 and they have stated that at these temperatures the stuff of mater breaks down into what is called a " quark-gluon plasma" in this state apparantly they are freed of their attraction to eachother. 

Further reading shows that the maximum theoretical temp is the Planck temerature which is 1.41679 x 1032 Kelvins.  I have no idea why there is a max. and I hope this is addressed in the answer on Sunday.
It seems things can get a lot hotter then they are right now but not much colder in comparison.

Aaron_Thomas, Fri, 23rd Mar 2012

Hi Athumbs - could you post a link to the article, I am at a loss to understand what a melting photon is! 

Be slightly wary of thinking of the Planck scale etc as physical maximums - more often they are levels at which present physical laws break down (ie we cannot describe it) rather than barriers to actual stuff.  The speed of light, abs zero - are actual limits; nothing faster, etc.  The planck scale is where quantum gravity is so important that present field theory collapses.

  From wikipage on Absolute Hot imatfaal, Fri, 23rd Mar 2012

This is incorrect Absolute hot is postulated at something like 1x10^32 C Negative energy exists as well. Alex, Wed, 5th Feb 2014

there does exist the upper limit of could be the temperature at the beginning of this is found to be TEN TO THE POWER OF THIRTY TWO KELVIN. Sudhakaran, Sat, 5th Sep 2015

I realize that this isn't exactly the same concept, but what if a photon was given so much energy that its wavelength approached the Planck length? KB, Mon, 4th Apr 2016

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