Heat lightning refers to distant flashes of lightning that occur on the horizon during hot weather and are not accompanied by thunder. It is incorrect to assume that the lightning is caused by the heat, but the term was coined because lightning without thunder usually takes place during the hot summer months. In contract to regular lightning, heat lightning is commonly seen over the ocean and is actually the remnant of a thunderstorm that is occurring a lot farther away.
What causes the lack of thunder with this type of lightning? In most cases there is thunder associated with the storm. It is just that you are too far away from the source to actually hear the sound of the thunder. The average speed of thunder is less than 10 miles an hour. The air is also composed of particles and densities that can affect how the sound travels and these will impact whether or not you hear any thunder even though you see the lightning. When the discharge from the sound of the thunder is affected by the density of the air, it appears to be silent lightning, which is exactly what heat lightning is.
The most intense thunderstorms can be seen up to 100 miles away when the conditions in the atmosphere are right. The lightning discharges illuminate the clouds, which makes this more visible at long distances away although the thunder may not be heard. Lightning storms also occur during the winter months during snowstorms and in these cases the sound of the wind may silence the sound of the thunder. In these cases, the storm cannot be called heat lightning.
The curvature of the earth also plays a role in whether or not you hear the thunder after you see lightning. Thunder bounces off the surface of the earth and the rumbling sound is part of this reflection. There can be voids in between the refraction and reflection of thunder and if you are in this void, then you will not be able to hear the sounds of the thunder associated with the storm. As a result you may be able to hear the thunder when you are quite far away from the storm, while others who are closer to the center cannot hear anything at all. The opposite this is also true, with those close to the storm being able to hear it and those far away only seeing the lightning.
So based on this information it is possible to conclude that there is no such thing as heat lightning if you mean that it is caused by excessive heat in the atmosphere. It should be termed silent lightning because of its association with seemingly non-existent thunder. These air mass thunderstorms are more likely to occur when the temperatures on the Earth's surface are warm and are almost always associated with rain.